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Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 184 (from laksa057)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa184.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://reference.wolfram.com/language/ref/LaplaceTransform.html\')), getAhrefsUnparsedNoserviceFromURL(\'https://reference.wolfram.com/language/ref/LaplaceTransform.html\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html","crawl_time":1776055361,"first_indexed_time":1410454304,"http_code":200,"src_unparsed":"com,wolfram!reference,\/language\/ref\/LaplaceTransform.html s443","src_root_hash":"3744487911316863784","history_drop_reason":null,"meta_title":"LaplaceTransform—Wolfram Documentation","meta_descriptions":["LaplaceTransform[f[t], t, s] gives the symbolic Laplace transform of f[t] in the variable t as F[s] in the variable s. LaplaceTransform[f[t], t, OverscriptBox[s, ^]] gives the numeric Laplace transform at the numerical value OverscriptBox[s, ^]. LaplaceTransform[f[t1, ..., tn], {t1, ..., tn}, {s1, ..., sn}] gives the multidimensional Laplace transform of f[t1, ..., tn]."],"attrs_boilerpipe_text":"See Also\nInverseLaplaceTransform\nBilateralLaplaceTransform\nFourierTransform\nUnilateralConvolve\nZTransform\nIntegrate\nPiecewise\nTransferFunctionModel\nMellinTransform\nAsymptotic\nFractionalD\nCaputoD\nRelated Guides\nFractional Calculus\nFourier Analysis\nGeneralized Functions\nSignal Transforms\nCalculus\nIntegral Transforms\nSignal Processing\nSummation Transforms\nTech Notes\nIntegral Transforms and Related Operations\nIntroduction to Fractional Calculus\nSee Also\nInverseLaplaceTransform\nBilateralLaplaceTransform\nFourierTransform\nUnilateralConvolve\nZTransform\nIntegrate\nPiecewise\nTransferFunctionModel\nMellinTransform\nAsymptotic\nFractionalD\nCaputoD\nRelated Guides\nFractional Calculus\nFourier Analysis\nGeneralized Functions\nSignal Transforms\nCalculus\nIntegral Transforms\nSignal Processing\nSummation Transforms\nTech Notes\nIntegral Transforms and Related Operations\nIntroduction to Fractional Calculus\nExamples\n \n \nBasic Examples\n \n \nScope\n \n \nBasic Uses\n \n \nElementary Functions\n \n \nSpecial Functions\n \n \nFormal Properties\n \n \nNumerical Evaluation\n \n \nFractional Calculus\n \n \nOptions\n \n \nAssumptions\n \n \nGenerateConditions\n \n \nPrincipal Value\n \n \nWorking Precision\n \n \nApplications\n \n \nOrdinary Differential Equations\n \n \nFractional Differential Equations\n \n \nEvaluation of Integrals\n \n \nOther Applications\n \n \nProperties & Relations\n \n \nPossible Issues\n \n \nNeat Examples\n \n \nLaplaceTransform\n[\nf\n[\nt\n]\n,\nt\n,\ns\n]\nCopy to clipboard.\nLaplaceTransform[f[t],t,s]\ngives the symbolic Laplace transform of\nf\n[\nt\n]\nin the variable\nt\nas\nF\n[\ns\n]\nin the variable\ns\n.\nLaplaceTransform\n[\nf\n[\nt\n]\n,\nt\n,\n]\nCopy to clipboard.\nLaplaceTransform[f[t],t,]\ngives the numeric Laplace transform at the numerical value\n.\nLaplaceTransform\n[\nf\n[\nt\n1\n,\n…\n,\nt\nn\n]\n,\n{\nt\n1\n,\n…\n,\nt\nn\n}\n,\n{\ns\n1\n,\n…\n,\ns\nn\n}\n]\nCopy to clipboard.\nLaplaceTransform[f[t1,…,tn],{t1,…,tn},{s1,…,sn}]\ngives the multidimensional Laplace transform of\nf\n[\nt\n1\n,\n…\n,\nt\nn\n]\n.\nDetails and Options\nLaplace transforms are typically used to transform differential and partial differential equations to algebraic equations, solve and then inverse transform back to a solution.\nLaplace transforms are also extensively used in control theory and signal processing as a way to represent and manipulate linear systems in the form of transfer functions and transfer matrices. The Laplace transform and its inverse are then a way to transform between the time domain and frequency domain.\nThe Laplace transform of a function\nis defined to be\n.\nThe multidimensional Laplace transform is given by\n.\nThe integral is computed using numerical methods if the third argument,\ns\n, is given a numerical value.\nThe asymptotic Laplace transform can be computed using\nAsymptotic\n.\nThe Laplace transform of\nexists only for complex values of\ns\nin a half-plane\n.\nThe lower limit of the integral is effectively taken to be\n, so that the Laplace transform of the Dirac delta function\nis equal to 1.\n»\nThe following options can be given:\nAccuracyGoal\nAutomatic\ndigits of absolute accuracy sought\nAssumptions\n$Assumptions\nassumptions to make about parameters\nGenerateConditions\nFalse\nwhether to generate answers that involve conditions on parameters\nMethod\nAutomatic\nmethod to use\nPerformanceGoal\n$PerformanceGoal\naspects of performance to optimize\nPrecisionGoal\nAutomatic\ndigits of precision sought\nPrincipalValue\nFalse\nwhether to find Cauchy principal value\nWorkingPrecision\nAutomatic\nthe precision used in internal computations\nUse\nGenerateConditions\n\n\"ConvergenceRegion\"\nto obtain the region of convergence for the Laplace transform.\nIn\nTraditionalForm\n,\nLaplaceTransform\nis output using\n.\n»\nExamples\nopen all\nclose all\nBasic Examples\n \n \n(4)\nSummary of the most common use cases\nScope\n \n \n(67)\nSurvey of the scope of standard use cases\nBasic Uses\n \n \n(4)\nElementary Functions\n \n \n(13)\nSpecial Functions\n \n \n(10)\nPiecewise Functions\n \n \n(9)\nPeriodic Functions\n \n \n(5)\nGeneralized Functions\n \n \n(5)\nMultivariate Functions\n \n \n(9)\nBivariate Laplace transform of a constant:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-l51olx\nDirect link to example\nOut[1]=\n1\nExponential function:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-iftr0v\nDirect link to example\nOut[1]=\n1\nPower function:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-xn08dr\nDirect link to example\nOut[1]=\n1\nBesselJ\n:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-pleo5d\nDirect link to example\nOut[1]=\n1\nSquare root:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-wilqmz\nDirect link to example\nOut[1]=\n1\nComposition of cosine and square root:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-m19040\nDirect link to example\nOut[1]=\n1\nLaplace transform of a multivariate power function:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-jvc1kh\nDirect link to example\nOut[1]=\n1\nCosine:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-w44hxj\nDirect link to example\nOut[1]=\n1\nLogarithm:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-nse2y4\nDirect link to example\nOut[1]=\n1\nFormal Properties\n \n \n(6)\nNumerical Evaluation\n \n \n(3)\nFractional Calculus\n \n \n(3)\nOptions\n \n \n(4)\nCommon values & functionality for each option\nAssumptions\n \n \n(1)\nSpecify the range for a parameter using\nAssumptions\n:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-fkqovc\nDirect link to example\nOut[1]=\n1\nGenerateConditions\n \n \n(1)\nUse\nGenerateConditions\n->\nTrue\nto get parameter conditions for when a result is valid:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-6pnei\nDirect link to example\nOut[1]=\n1\nPrincipal Value\n \n \n(1)\nWorking Precision\n \n \n(1)\nApplications\n \n \n(12)\nSample problems that can be solved with this function\nOrdinary Differential Equations\n \n \n(5)\nFractional Differential Equations\n \n \n(3)\nSolve a fractional-order differential equation using Laplace transforms:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-5yhytr\nDirect link to example\nOut[1]=\n1\nSolve for the Laplace transform:\nCopy to clipboard.\nIn[2]:=\n2\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-8g7evs\nDirect link to example\nOut[2]=\n2\nFind the inverse transform:\nCopy to clipboard.\nIn[3]:=\n3\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-203z1c\nDirect link to example\nOut[3]=\n3\nPlot the solution:\nCopy to clipboard.\nIn[4]:=\n4\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-oepin2\nDirect link to example\nOut[4]=\n4\nFind the solution directly using\nDSolve\n:\nCopy to clipboard.\nIn[5]:=\n5\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-iiovhh\nDirect link to example\nOut[5]=\n5\nSolve the following fractional integro-differential equation:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-sboa2x\nDirect link to example\nSolve for the Laplace transform:\nCopy to clipboard.\nIn[2]:=\n2\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-6hxn7n\nDirect link to example\nOut[2]=\n2\nFind the inverse transform:\nCopy to clipboard.\nIn[3]:=\n3\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-ik5ypj\nDirect link to example\nOut[3]=\n3\nFind the solution directly using\nDSolve\n:\nCopy to clipboard.\nIn[4]:=\n4\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-hvse9n\nDirect link to example\nOut[4]=\n4\nThe following equation describes a fractional harmonic oscillator of order 1.9:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-usm5e4\nDirect link to example\nSolve for the Laplace transform:\nCopy to clipboard.\nIn[2]:=\n2\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-zsldvm\nDirect link to example\nOut[2]=\n2\nFind the inverse transform:\nCopy to clipboard.\nIn[3]:=\n3\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-x8hdfh\nDirect link to example\nOut[3]=\n3\nPlot the solution:\nCopy to clipboard.\nIn[4]:=\n4\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-msydxu\nDirect link to example\nOut[4]=\n4\nFind the solution directly using\nDSolve\n:\nCopy to clipboard.\nIn[5]:=\n5\nhttps:\/\/wolfram.com\/xid\/0h2pdw27n2-ogzgeo\nDirect link to example\nOut[5]=\n5\nEvaluation of Integrals\n \n \n(2)\nOther Applications\n \n \n(2)\nProperties & Relations\n \n \n(3)\nProperties of the function, and connections to other functions\nPossible Issues\n \n \n(1)\nCommon pitfalls and unexpected behavior\nNeat Examples\n \n \n(2)\nSurprising or curious use cases\nWolfram Research (1999), LaplaceTransform, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html (updated 2023).\nCopy to clipboard.\nWolfram Research (1999), LaplaceTransform, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html (updated 2023).\nText\nWolfram Research (1999), LaplaceTransform, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html (updated 2023).\nCopy to clipboard.\nWolfram Research (1999), LaplaceTransform, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html (updated 2023).\nCMS\nWolfram Language. 1999. \"LaplaceTransform.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html.\nCopy to clipboard.\nWolfram Language. 1999. \"LaplaceTransform.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html.\nAPA\nWolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html\nCopy to clipboard.\nWolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html\nBibTeX\n@misc{reference.wolfram_2025_laplacetransform, author=\"Wolfram Research\", title=\"{LaplaceTransform}\", year=\"2023\", howpublished=\"\\url{https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html}\", note=[Accessed: 12-April-2026]}\nCopy to clipboard.\n@misc{reference.wolfram_2025_laplacetransform, author=\"Wolfram Research\", title=\"{LaplaceTransform}\", year=\"2023\", howpublished=\"\\url{https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html}\", note=[Accessed: 12-April-2026]}\nBibLaTeX\n@online{reference.wolfram_2025_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html}, note=[Accessed: 12-April-2026]}\nCopy to clipboard.\n@online{reference.wolfram_2025_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html}, note=[Accessed: 12-April-2026]}","attrs_markdown":"#\n[Products](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n\n- [Wolfram\\|One The definitive Wolfram Language and notebook experience](https:\/\/www.wolfram.com\/wolfram-one\/?source=nav)\n- [Mathematica The original technical computing environment](https:\/\/www.wolfram.com\/mathematica\/?source=nav)\n- [Notebook Assistant + LLM Kit All-in-one AI assistance for your Wolfram experience](https:\/\/www.wolfram.com\/notebook-assistant-llm-kit\/?source=nav)\n\n- [Compute Services](https:\/\/www.wolfram.com\/compute-services\/?source=nav)\n- [System Modeler](https:\/\/www.wolfram.com\/system-modeler\/?source=nav)\n- [Finance Platform](https:\/\/www.wolfram.com\/finance-platform\/?source=nav)\n- [Wolfram\\|Alpha Notebook Edition](https:\/\/www.wolfram.com\/wolfram-alpha-notebook-edition\/?source=nav)\n- [Application 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[UnilateralConvolve](https:\/\/reference.wolfram.com\/language\/ref\/UnilateralConvolve.html)\n  - [ZTransform](https:\/\/reference.wolfram.com\/language\/ref\/ZTransform.html)\n  - [Integrate](https:\/\/reference.wolfram.com\/language\/ref\/Integrate.html)\n  - [Piecewise](https:\/\/reference.wolfram.com\/language\/ref\/Piecewise.html)\n  - [TransferFunctionModel](https:\/\/reference.wolfram.com\/language\/ref\/TransferFunctionModel.html)\n  - [MellinTransform](https:\/\/reference.wolfram.com\/language\/ref\/MellinTransform.html)\n  - [Asymptotic](https:\/\/reference.wolfram.com\/language\/ref\/Asymptotic.html)\n  - [FractionalD](https:\/\/reference.wolfram.com\/language\/ref\/FractionalD.html)\n  - [CaputoD](https:\/\/reference.wolfram.com\/language\/ref\/CaputoD.html)\n- [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n  - [Fractional Calculus](https:\/\/reference.wolfram.com\/language\/guide\/FractionalCalculus.html)\n  - [Fourier Analysis](https:\/\/reference.wolfram.com\/language\/guide\/FourierAnalysis.html)\n  - [Generalized Functions](https:\/\/reference.wolfram.com\/language\/guide\/GeneralizedFunctions.html)\n  - [Signal Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SignalTransforms.html)\n  - [Calculus](https:\/\/reference.wolfram.com\/language\/guide\/Calculus.html)\n  - [Integral Transforms](https:\/\/reference.wolfram.com\/language\/guide\/IntegralTransforms.html)\n  - [Signal Processing](https:\/\/reference.wolfram.com\/language\/guide\/SignalProcessing.html)\n  - [Summation Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SummationTransforms.html)\n- [Tech Notes](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n  - [Integral Transforms and Related Operations](https:\/\/reference.wolfram.com\/language\/tutorial\/Calculus.html#26017)\n  - [Introduction to Fractional Calculus](https:\/\/reference.wolfram.com\/language\/tutorial\/FractionalCalculus.html)\n- - [See Also](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n    - [InverseLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/InverseLaplaceTransform.html)\n    - [BilateralLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/BilateralLaplaceTransform.html)\n    - [FourierTransform](https:\/\/reference.wolfram.com\/language\/ref\/FourierTransform.html)\n    - [UnilateralConvolve](https:\/\/reference.wolfram.com\/language\/ref\/UnilateralConvolve.html)\n    - [ZTransform](https:\/\/reference.wolfram.com\/language\/ref\/ZTransform.html)\n    - [Integrate](https:\/\/reference.wolfram.com\/language\/ref\/Integrate.html)\n    - [Piecewise](https:\/\/reference.wolfram.com\/language\/ref\/Piecewise.html)\n    - [TransferFunctionModel](https:\/\/reference.wolfram.com\/language\/ref\/TransferFunctionModel.html)\n    - [MellinTransform](https:\/\/reference.wolfram.com\/language\/ref\/MellinTransform.html)\n    - [Asymptotic](https:\/\/reference.wolfram.com\/language\/ref\/Asymptotic.html)\n    - [FractionalD](https:\/\/reference.wolfram.com\/language\/ref\/FractionalD.html)\n    - [CaputoD](https:\/\/reference.wolfram.com\/language\/ref\/CaputoD.html)\n  - [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n    - [Fractional Calculus](https:\/\/reference.wolfram.com\/language\/guide\/FractionalCalculus.html)\n    - [Fourier Analysis](https:\/\/reference.wolfram.com\/language\/guide\/FourierAnalysis.html)\n    - [Generalized Functions](https:\/\/reference.wolfram.com\/language\/guide\/GeneralizedFunctions.html)\n    - [Signal Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SignalTransforms.html)\n    - [Calculus](https:\/\/reference.wolfram.com\/language\/guide\/Calculus.html)\n    - [Integral Transforms](https:\/\/reference.wolfram.com\/language\/guide\/IntegralTransforms.html)\n    - [Signal Processing](https:\/\/reference.wolfram.com\/language\/guide\/SignalProcessing.html)\n    - [Summation Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SummationTransforms.html)\n  - [Tech Notes](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n    - [Integral Transforms and Related Operations](https:\/\/reference.wolfram.com\/language\/tutorial\/Calculus.html#26017)\n    - [Introduction to Fractional Calculus](https:\/\/reference.wolfram.com\/language\/tutorial\/FractionalCalculus.html)\n\n[LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\\[f\\[t\\],t,s\\]\n\ngives the symbolic Laplace transform of f\\[t\\] in the variable t as F\\[s\\] in the variable s.\n\n[LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\\[f\\[t\\],t,![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/28OverscriptBox.png)\\]\n\ngives the numeric Laplace transform at the numerical value ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/29OverscriptBox.png).\n\n[LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\\[f\\[t1,…,tn\\],{t1,…,tn},{s1,…,sn}\\]\n\ngives the multidimensional Laplace transform of f\\[t1,…,tn\\].\n\nDetails and Options\n\n![Details and Options](https:\/\/reference.wolfram.com\/language-assets-v2\/img\/DetailsOptions-gray.png) ![Details and Options](https:\/\/reference.wolfram.com\/language-assets-v2\/img\/DetailsOptions-hover.png)\n\nExamples\n\nBasic Examples\n\nScope\n\nBasic Uses\n\nElementary Functions\n\nSpecial Functions\n\n![Show More](https:\/\/reference.wolfram.com\/language-assets-v2\/img\/Options-hamburger-gray.svg) ![Show More](https:\/\/reference.wolfram.com\/language-assets-v2\/img\/Options-hamburger-hover.svg)\n\nPiecewise Functions\n\nPeriodic Functions\n\nGeneralized Functions\n\nMultivariate Functions\n\nFormal Properties\n\nNumerical Evaluation\n\nFractional Calculus\n\nOptions\n\nAssumptions\n\nGenerateConditions\n\nPrincipal Value\n\nWorking Precision\n\nApplications\n\nOrdinary Differential Equations\n\nFractional Differential Equations\n\nEvaluation of Integrals\n\nOther Applications\n\nProperties & Relations\n\nPossible Issues\n\nNeat Examples\n\nSee Also\n\nTech Notes\n\nRelated Guides\n\nRelated Links\n\nHistory\n\nCite this Page\n\nBUILT-IN SYMBOL\n\n- [See Also](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n  - [InverseLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/InverseLaplaceTransform.html)\n  - [BilateralLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/BilateralLaplaceTransform.html)\n  - [FourierTransform](https:\/\/reference.wolfram.com\/language\/ref\/FourierTransform.html)\n  - [UnilateralConvolve](https:\/\/reference.wolfram.com\/language\/ref\/UnilateralConvolve.html)\n  - [ZTransform](https:\/\/reference.wolfram.com\/language\/ref\/ZTransform.html)\n  - [Integrate](https:\/\/reference.wolfram.com\/language\/ref\/Integrate.html)\n  - [Piecewise](https:\/\/reference.wolfram.com\/language\/ref\/Piecewise.html)\n  - [TransferFunctionModel](https:\/\/reference.wolfram.com\/language\/ref\/TransferFunctionModel.html)\n  - [MellinTransform](https:\/\/reference.wolfram.com\/language\/ref\/MellinTransform.html)\n  - [Asymptotic](https:\/\/reference.wolfram.com\/language\/ref\/Asymptotic.html)\n  - [FractionalD](https:\/\/reference.wolfram.com\/language\/ref\/FractionalD.html)\n  - [CaputoD](https:\/\/reference.wolfram.com\/language\/ref\/CaputoD.html)\n- [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n  - [Fractional Calculus](https:\/\/reference.wolfram.com\/language\/guide\/FractionalCalculus.html)\n  - [Fourier Analysis](https:\/\/reference.wolfram.com\/language\/guide\/FourierAnalysis.html)\n  - [Generalized Functions](https:\/\/reference.wolfram.com\/language\/guide\/GeneralizedFunctions.html)\n  - [Signal Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SignalTransforms.html)\n  - [Calculus](https:\/\/reference.wolfram.com\/language\/guide\/Calculus.html)\n  - [Integral Transforms](https:\/\/reference.wolfram.com\/language\/guide\/IntegralTransforms.html)\n  - [Signal Processing](https:\/\/reference.wolfram.com\/language\/guide\/SignalProcessing.html)\n  - [Summation Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SummationTransforms.html)\n- [Tech Notes](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n  - [Integral Transforms and Related Operations](https:\/\/reference.wolfram.com\/language\/tutorial\/Calculus.html#26017)\n  - [Introduction to Fractional Calculus](https:\/\/reference.wolfram.com\/language\/tutorial\/FractionalCalculus.html)\n- - [See Also](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n    - [InverseLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/InverseLaplaceTransform.html)\n    - [BilateralLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/BilateralLaplaceTransform.html)\n    - [FourierTransform](https:\/\/reference.wolfram.com\/language\/ref\/FourierTransform.html)\n    - [UnilateralConvolve](https:\/\/reference.wolfram.com\/language\/ref\/UnilateralConvolve.html)\n    - [ZTransform](https:\/\/reference.wolfram.com\/language\/ref\/ZTransform.html)\n    - [Integrate](https:\/\/reference.wolfram.com\/language\/ref\/Integrate.html)\n    - [Piecewise](https:\/\/reference.wolfram.com\/language\/ref\/Piecewise.html)\n    - [TransferFunctionModel](https:\/\/reference.wolfram.com\/language\/ref\/TransferFunctionModel.html)\n    - [MellinTransform](https:\/\/reference.wolfram.com\/language\/ref\/MellinTransform.html)\n    - [Asymptotic](https:\/\/reference.wolfram.com\/language\/ref\/Asymptotic.html)\n    - [FractionalD](https:\/\/reference.wolfram.com\/language\/ref\/FractionalD.html)\n    - [CaputoD](https:\/\/reference.wolfram.com\/language\/ref\/CaputoD.html)\n  - [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n    - [Fractional Calculus](https:\/\/reference.wolfram.com\/language\/guide\/FractionalCalculus.html)\n    - [Fourier Analysis](https:\/\/reference.wolfram.com\/language\/guide\/FourierAnalysis.html)\n    - [Generalized Functions](https:\/\/reference.wolfram.com\/language\/guide\/GeneralizedFunctions.html)\n    - [Signal Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SignalTransforms.html)\n    - [Calculus](https:\/\/reference.wolfram.com\/language\/guide\/Calculus.html)\n    - [Integral Transforms](https:\/\/reference.wolfram.com\/language\/guide\/IntegralTransforms.html)\n    - [Signal Processing](https:\/\/reference.wolfram.com\/language\/guide\/SignalProcessing.html)\n    - [Summation Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SummationTransforms.html)\n  - [Tech Notes](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n    - [Integral Transforms and Related Operations](https:\/\/reference.wolfram.com\/language\/tutorial\/Calculus.html#26017)\n    - [Introduction to Fractional Calculus](https:\/\/reference.wolfram.com\/language\/tutorial\/FractionalCalculus.html)\n# LaplaceTransformCopy to clipboard. ✖ `LaplaceTransform`\n\n[LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\\[f\\[t\\],t,s\\]\n\nCopy to clipboard.\n\n✖\n\n`LaplaceTransform[f[t],t,s]`\n\ngives the symbolic Laplace transform of f\\[t\\] in the variable t as F\\[s\\] in the variable s.\n\n[LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\\[f\\[t\\],t,![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/1OverscriptBox.png)\\]\n\nCopy to clipboard.\n\n✖\n\n`LaplaceTransform[f[t],t,]`\n\ngives the numeric Laplace transform at the numerical value ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/2OverscriptBox.png).\n\n[LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\\[f\\[t1,…,tn\\],{t1,…,tn},{s1,…,sn}\\]\n\nCopy to clipboard.\n\n✖\n\n`LaplaceTransform[f[t1,…,tn],{t1,…,tn},{s1,…,sn}]`\n\ngives the multidimensional Laplace transform of f\\[t1,…,tn\\].\n\n# Details and Options\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/details_1.png) ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/details_2.png)\n\n- Laplace transforms are typically used to transform differential and partial differential equations to algebraic equations, solve and then inverse transform back to a solution.\n- Laplace transforms are also extensively used in control theory and signal processing as a way to represent and manipulate linear systems in the form of transfer functions and transfer matrices. The Laplace transform and its inverse are then a way to transform between the time domain and frequency domain.\n- The Laplace transform of a function ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/3.png) is defined to be ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/4.png).\n- The multidimensional Laplace transform is given by ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/5.png).\n- The integral is computed using numerical methods if the third argument, s, is given a numerical value.\n- The asymptotic Laplace transform can be computed using [Asymptotic](https:\/\/reference.wolfram.com\/language\/ref\/Asymptotic.html).\n- The Laplace transform of ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/6.png) exists only for complex values of s in a half-plane ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/7.png).\n- The lower limit of the integral is effectively taken to be ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/9.png), so that the Laplace transform of the Dirac delta function ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/10.png) is equal to 1.  [»](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html#427361964)\n- The following options can be given:\n- |   |   |   |   |\n  |---|---|---|---|\n  |   | [AccuracyGoal](https:\/\/reference.wolfram.com\/language\/ref\/AccuracyGoal.html) | [Automatic](https:\/\/reference.wolfram.com\/language\/ref\/Automatic.html) | digits of absolute accuracy sought |\n  |   | [Assumptions](https:\/\/reference.wolfram.com\/language\/ref\/Assumptions.html) | [\\$Assumptions](https:\/\/reference.wolfram.com\/language\/ref\/$Assumptions.html) | assumptions to make about parameters |\n  |   | [GenerateConditions](https:\/\/reference.wolfram.com\/language\/ref\/GenerateConditions.html) | [False](https:\/\/reference.wolfram.com\/language\/ref\/False.html) | whether to generate answers that involve conditions on parameters |\n  |   | [Method](https:\/\/reference.wolfram.com\/language\/ref\/Method.html) | [Automatic](https:\/\/reference.wolfram.com\/language\/ref\/Automatic.html) | method to use |\n  |   | [PerformanceGoal](https:\/\/reference.wolfram.com\/language\/ref\/PerformanceGoal.html) | [\\$PerformanceGoal](https:\/\/reference.wolfram.com\/language\/ref\/$PerformanceGoal.html) | aspects of performance to optimize |\n  |   | [PrecisionGoal](https:\/\/reference.wolfram.com\/language\/ref\/PrecisionGoal.html) | [Automatic](https:\/\/reference.wolfram.com\/language\/ref\/Automatic.html) | digits of precision sought |\n  |   | [PrincipalValue](https:\/\/reference.wolfram.com\/language\/ref\/PrincipalValue.html) | [False](https:\/\/reference.wolfram.com\/language\/ref\/False.html) | whether to find Cauchy principal value |\n  |   | [WorkingPrecision](https:\/\/reference.wolfram.com\/language\/ref\/WorkingPrecision.html) | [Automatic](https:\/\/reference.wolfram.com\/language\/ref\/Automatic.html) | the precision used in internal computations |\n- Use [GenerateConditions](https:\/\/reference.wolfram.com\/language\/ref\/GenerateConditions.html)\"ConvergenceRegion\" to obtain the region of convergence for the Laplace transform.\n- In [TraditionalForm](https:\/\/reference.wolfram.com\/language\/ref\/TraditionalForm.html), [LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html) is output using ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/11.png). [»](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html#2079194920)\n\n# Examples\nopen all close all\n\n## Basic Examples  (4)Summary of the most common use cases\nCompute the Laplace transform of a function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_1.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-52saw3`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_1.png)\n\nDefine a piecewise function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_2.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-fl1sjt`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_3.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-60cdu`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_2.png)\n\nCompute its Laplace transform:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_4.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-7hhqvo`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_3.png)\n\nCompute the transform at a single point:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_5.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-fqa72t`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_4.png)\n\nCompute the Laplace transform of a multivariate function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_6.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-0ewuhr`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_5.png)\n\nDefine a multivariate piecewise function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_7.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-fom62v`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_8.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-69gxmn`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_6.png)\n\nCompute its Laplace transform:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_9.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-eqq3xl`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_7.png)\n\n## Scope  (67)Survey of the scope of standard use cases\n### Basic Uses  (4)\nLaplace transform of a function for a symbolic parameter s:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_10.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-vlts3p`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_8.png)\n\nLaplace transforms of trigonometric functions:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_11.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-qf9a9`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_9.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_12.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-bargvx`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_10.png)\n\nEvaluate the Laplace transform for a numerical value of the parameter s:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_13.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-4ycwlj`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_11.png)\n\n[TraditionalForm](https:\/\/reference.wolfram.com\/language\/ref\/TraditionalForm.html) formatting:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_14.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-tpw3x2`\n\nDirect link to example\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_12.png)\n\n### Elementary Functions  (13)\nLaplace transform of a power function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_15.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-bxq0e5`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_13.png)\n\nSquare root function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_16.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-l3vqf0`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_14.png)\n\nLaplace transforms of polynomials:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_17.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-hevqei`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_15.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_18.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-d05omh`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_16.png)\n\nExponential function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_19.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-kvdxn3`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_17.png)\n\nProduct of an exponential and a linear function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_20.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-104rjl`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_18.png)\n\nExpressions involving trigonometric functions:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_21.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-tqq1h2`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_19.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_22.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-gd0kyr`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_20.png)\n\nExpressions involving hyperbolic functions:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_23.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-i8tzxa`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_21.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_24.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-3t5n0r`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_22.png)\n\nRatio of an exponential and a linear function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_25.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-gbulo8`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_23.png)\n\nRatio of sine and linear functions:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_26.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-bsm8py`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_24.png)\n\nComposition of elementary functions:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_27.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-s9a3k`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_25.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_28.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-lwqt7u`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_26.png)\n\nLogarithmic function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_29.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-5bwuuf`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_27.png)\n\nProduct of logarithmic and power functions:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_30.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-p2nd7m`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_28.png)\n\nSquare of a logarithmic function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_31.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-j4w84x`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_29.png)\n\n### Special Functions  (10)\nLaplace transform of error and square root functions composition:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_32.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-k0nw19`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_30.png)\n\nBessel functions:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_33.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-9irs5p`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_31.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_34.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-2gxc7`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_32.png)\n\nProducts involving Bessel functions:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_35.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-pbgdhr`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_33.png)\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_36.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-c04r43`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_34.png)\n\nSine integral function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_37.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-q8ehy8`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_35.png)\n\nLaguerre polynomials:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_38.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-xzqhic`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_36.png)\n\nAiry function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_39.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-hso455`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_37.png)\n\nChebyshev polynomial:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_40.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-rt1kwi`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_38.png)\n\nStruve function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_41.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-p1ki8t`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_39.png)\n\nFresnel function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_42.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-5bn4go`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_40.png)\n\nGamma function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_43.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-2yz0oy`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_41.png)\n\nHypergeometric function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_44.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-4u6kzh`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_42.png)\n\n### Piecewise Functions  (9)\nLaplace transform of a piecewise function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_45.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-h4lxo8`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_46.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-b05sk8`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_43.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_47.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-ruge2u`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_44.png)\n\nRestriction of a sine function to a half-period:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_48.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-2hml80`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_49.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-iwlcqb`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_45.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_50.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-g1bigg`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_46.png)\n\nExponential function with a left cutoff:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_51.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-mw0vs1`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_52.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-iyy5ib`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_47.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_53.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-hj9cvx`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_48.png)\n\nTriangular function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_54.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-kor8nt`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_55.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-09ydxm`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_49.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_56.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-ldq8my`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_50.png)\n\nPolynomial function with a left cutoff:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_57.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-499t0o`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_58.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-e39213`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_51.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_59.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-7yy7ev`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_52.png)\n\nRamp:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_60.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-ydoxba`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_53.png)\n\n[UnitStep](https:\/\/reference.wolfram.com\/language\/ref\/UnitStep.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_61.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-sqqyn7`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_54.png)\n\nProduct of [UnitStep](https:\/\/reference.wolfram.com\/language\/ref\/UnitStep.html) and cosine functions:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_62.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-48rqn1`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_55.png)\n\nLaplace transform of [Floor](https:\/\/reference.wolfram.com\/language\/ref\/Floor.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_63.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-6akvqj`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_56.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_64.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-bmflvc`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_57.png)\n\n### Periodic Functions  (5)\nLaplace transform of [SquareWave](https:\/\/reference.wolfram.com\/language\/ref\/SquareWave.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_65.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-7ktv9h`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_58.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_66.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-ycjry8`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_59.png)\n\n[TriangleWave](https:\/\/reference.wolfram.com\/language\/ref\/TriangleWave.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_67.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-4qy61o`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_60.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_68.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-2avn8l`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_61.png)\n\n[SawtoothWave](https:\/\/reference.wolfram.com\/language\/ref\/SawtoothWave.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_69.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-u745mq`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_62.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_70.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-emltg7`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_63.png)\n\nFull-wave-rectified function with period ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/12.png):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_71.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-nb3raf`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_64.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_72.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-byq9w3`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_65.png)\n\nRectified wave:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_73.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-no882i`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_74.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-fig26d`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_66.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_75.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-zqhvt8`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_67.png)\n\n### Generalized Functions  (5)\nLaplace transform of [HeavisideTheta](https:\/\/reference.wolfram.com\/language\/ref\/HeavisideTheta.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_76.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-pg20cq`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_68.png)\n\n[DiracDelta](https:\/\/reference.wolfram.com\/language\/ref\/DiracDelta.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_77.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-yvqfg2`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_69.png)\n\nDerivative of [DiracDelta](https:\/\/reference.wolfram.com\/language\/ref\/DiracDelta.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_78.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-f8njof`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_70.png)\n\n[HeavisideLambda](https:\/\/reference.wolfram.com\/language\/ref\/HeavisideLambda.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_79.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-uw6trd`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_71.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_80.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-1faykd`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_72.png)\n\n[HeavisidePi](https:\/\/reference.wolfram.com\/language\/ref\/HeavisidePi.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_81.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-uv2j6r`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_73.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_82.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-2m3had`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_74.png)\n\n### Multivariate Functions  (9)\nBivariate Laplace transform of a constant:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_83.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-l51olx`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_75.png)\n\nExponential function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_84.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-iftr0v`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_76.png)\n\nPower function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_85.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-xn08dr`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_77.png)\n\n[BesselJ](https:\/\/reference.wolfram.com\/language\/ref\/BesselJ.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_86.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-pleo5d`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_78.png)\n\nSquare root:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_87.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-wilqmz`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_79.png)\n\nComposition of cosine and square root:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_88.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-m19040`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_80.png)\n\nLaplace transform of a multivariate power function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_89.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-jvc1kh`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_81.png)\n\nCosine:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_90.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-w44hxj`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_82.png)\n\nLogarithm:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_91.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-nse2y4`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_83.png)\n\n### Formal Properties  (6)\nThe Laplace transform is a linear operator:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_92.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-sv38rg`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_84.png)\n\nLaplace transform of ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/13.png) is the Laplace transform of ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/14.png) evaluated at ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/15.png):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_93.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-3p91yz`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_85.png)\n\nLaplace transform of a first-order derivative:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_94.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-s0daap`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_86.png)\n\nLaplace transform of a second-order derivative:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_95.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-uwnfg0`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_87.png)\n\nLaplace transform of a product with monomials:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_96.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-kfzvq4`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_88.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_97.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-btnn48`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_89.png)\n\nLaplace transform threads itself over equations:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_98.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-kyrnxk`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_90.png)\n\n### Numerical Evaluation  (3)\nCalculate the Laplace transform at a single point:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_99.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-fhhyve`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_91.png)\n\nAlternatively, calculate the Laplace transform symbolically:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_100.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-1ten3m`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_92.png)\n\nThen evaluate it for specific value of ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/16.png):\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_101.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-bmvmnm`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_93.png)\n\nPlot the Laplace transform using numerical values only:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_102.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-lyv72q`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_103.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-ir3wks`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_94.png)\n\nFor some functions, the Laplace transform cannot be evaluated symbolically:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_104.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-gcfdhk`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_95.png)\n\nEvaluate the Laplace transform numerically and plot it:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_105.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-bomtml`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_106.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-cs3x94`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_96.png)\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_107.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-nd2svs`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_97.png)\n\nCalculate a multivariate Laplace transform at a single point in the plane:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_108.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-wqufw9`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_98.png)\n\n### Fractional Calculus  (3)\nLaplace transform of the [MittagLefflerE](https:\/\/reference.wolfram.com\/language\/ref\/MittagLefflerE.html) functions:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_109.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-9fi54c`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_99.png)\n\n[ComplexPlot](https:\/\/reference.wolfram.com\/language\/ref\/ComplexPlot.html) in the ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/17.png)\\-domain:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_110.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-3nzaga`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_100.png)\n\nInverse Laplace transform to the time domain:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_111.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-ox1krh`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_101.png)\n\nLaplace transform of the [MittagLefflerE](https:\/\/reference.wolfram.com\/language\/ref\/MittagLefflerE.html) functions involving parameters:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_112.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-s6e1zo`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_102.png)\n\nInverse Laplace transform to the time domain:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_113.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-2xte5w`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_103.png)\n\nLaplace transform of the [CaputoD](https:\/\/reference.wolfram.com\/language\/ref\/CaputoD.html) fractional derivative:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_114.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-dyaoq6`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_104.png)\n\nApply to sine function:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_115.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-dvqm5x`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_105.png)\n\nCompare this with the [LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html) of the [CaputoD](https:\/\/reference.wolfram.com\/language\/ref\/CaputoD.html) derivative of the sine function:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_116.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-p7qk4m`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_106.png)\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_117.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-csb1rs`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_107.png)\n\n## Options  (4)Common values & functionality for each option\n### Assumptions  (1)\nSpecify the range for a parameter using [Assumptions](https:\/\/reference.wolfram.com\/language\/ref\/Assumptions.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_118.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-fkqovc`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_108.png)\n\n### GenerateConditions  (1)\nUse [GenerateConditions](https:\/\/reference.wolfram.com\/language\/ref\/GenerateConditions.html)\\-\\>[True](https:\/\/reference.wolfram.com\/language\/ref\/True.html) to get parameter conditions for when a result is valid:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_119.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-6pnei`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_109.png)\n\n### Principal Value  (1)\nThe Laplace transform of the following function is not defined due to the singularity at ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/18.png):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_120.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-byox9b`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_110.png)\n\nUse [PrincipalValue](https:\/\/reference.wolfram.com\/language\/ref\/PrincipalValue.html) to obtain the Cauchy principal value for the integral:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_121.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-ly80t6`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_111.png)\n\n### Working Precision  (1)\nUse [WorkingPrecision](https:\/\/reference.wolfram.com\/language\/ref\/WorkingPrecision.html) to obtain a result with arbitrary precision:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_122.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-cm1yhe`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_112.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_123.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-jp0k2`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_113.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_124.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-gylrkk`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_114.png)\n\n## Applications  (12)Sample problems that can be solved with this function\n### Ordinary Differential Equations  (5)\nSolve a differential equation using Laplace transforms:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_125.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-z3evmw`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_115.png)\n\nSolve for the Laplace transform:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_126.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-spc3fq`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_116.png)\n\nFind the inverse transform:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_127.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-8kjf6g`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_117.png)\n\nPlot the solution:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_128.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-kwb15a`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_118.png)\n\nFind the solution directly using [DSolve](https:\/\/reference.wolfram.com\/language\/ref\/DSolve.html):\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_129.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-w27252`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_119.png)\n\nSolve the following differential equation:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_130.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-q5z19`\n\nDirect link to example\n\nSolve for the Laplace transform:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_131.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-suvmxo`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_120.png)\n\nFind the inverse transform:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_132.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-jycu84`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_121.png)\n\nPlot the solution:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_133.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-61tbxr`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_122.png)\n\nSolve an RL circuit to find the current ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/19.png):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_134.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-kzueyy`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_123.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_135.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-xt58sb`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_124.png)\n\nVerify with [DSolveValue](https:\/\/reference.wolfram.com\/language\/ref\/DSolveValue.html):\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_136.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-z2p4b4`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_125.png)\n\nGreen's function for an RL circuit:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_137.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-kllajx`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_126.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_138.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-3r61x2`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_127.png)\n\nUse the Green's function to solve the RL circuit:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_139.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-10tpdo`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_128.png)\n\nSolve a system of ODEs:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_140.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-zm5oii`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_129.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_141.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-5tqdr8`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_130.png)\n\n### Fractional Differential Equations  (3)\nSolve a fractional-order differential equation using Laplace transforms:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_142.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-5yhytr`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_131.png)\n\nSolve for the Laplace transform:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_143.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-8g7evs`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_132.png)\n\nFind the inverse transform:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_144.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-203z1c`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_133.png)\n\nPlot the solution:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_145.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-oepin2`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_134.png)\n\nFind the solution directly using [DSolve](https:\/\/reference.wolfram.com\/language\/ref\/DSolve.html):\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_146.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-iiovhh`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_135.png)\n\nSolve the following fractional integro-differential equation:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_147.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-sboa2x`\n\nDirect link to example\n\nSolve for the Laplace transform:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_148.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-6hxn7n`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_136.png)\n\nFind the inverse transform:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_149.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-ik5ypj`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_137.png)\n\nFind the solution directly using [DSolve](https:\/\/reference.wolfram.com\/language\/ref\/DSolve.html):\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_150.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-hvse9n`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_138.png)\n\nThe following equation describes a fractional harmonic oscillator of order 1.9:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_151.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-usm5e4`\n\nDirect link to example\n\nSolve for the Laplace transform:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_152.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-zsldvm`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_139.png)\n\nFind the inverse transform:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_153.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-x8hdfh`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_140.png)\n\nPlot the solution:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_154.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-msydxu`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_141.png)\n\nFind the solution directly using [DSolve](https:\/\/reference.wolfram.com\/language\/ref\/DSolve.html):\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_155.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-ogzgeo`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_142.png)\n\n### Evaluation of Integrals  (2)\nCalculate the following integral:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_156.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-79kl21`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_143.png)\n\nCompute the Laplace transform and interchange the order of Laplace transform and integration:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_157.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-t5iy5u`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_144.png)\n\nPerform the integration over ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/20.png):\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_158.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-l6recg`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_145.png)\n\nUse [InverseLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/InverseLaplaceTransform.html) to obtain the original integral:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_159.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-4fo1vb`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_146.png)\n\nVerify the result:\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_160.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-feq5fl`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_147.png)\n\nIntegral involving the Bessel function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_161.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-9dc9dg`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_148.png)\n\nPerform a change of variables ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/21.png) and introduce an auxiliary variable ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/22.png):\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_162.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-7gvszw`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_149.png)\n\nApply the Laplace transform and interchange the order of Laplace transform and integration:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_163.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-7aox3f`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_150.png)\n\nPerform the integration over ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/23.png):\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_164.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-f0ke5v`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_151.png)\n\nUse [InverseLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/InverseLaplaceTransform.html) to obtain ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/24.png):\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_165.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-kks51`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_152.png)\n\nThe original integral equals ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/25.png):\n\nCopy to clipboard.\n\nIn\\[6\\]:=6\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_166.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-9h7nq4`\n\nDirect link to example\n\nOut\\[6\\]=6\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_153.png)\n\nVerify the result:\n\nCopy to clipboard.\n\nIn\\[7\\]:=7\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_167.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-v81car`\n\nDirect link to example\n\nOut\\[7\\]=7\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_154.png)\n\n### Other Applications  (2)\nCompute a Laplace transform using a series expansion:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_168.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-wt994`\n\nDirect link to example\n\nThe odd coefficients vanish:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_169.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-699ns9`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_155.png)\n\nThe transformed series can be summed using [Regularization](https:\/\/reference.wolfram.com\/language\/ref\/Regularization.html):\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_170.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-eivhga`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_156.png)\n\nVerify the result directly using [LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html):\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_171.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-dvmuze`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_157.png)\n\nLaplace transform of [Sinc](https:\/\/reference.wolfram.com\/language\/ref\/Sinc.html) using series expansions:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_172.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-8z42as`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_173.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-19re7`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_158.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_174.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-fr6xz5`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_159.png)\n\nOdd coefficients vanish:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_175.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-cv31y4`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_160.png)\n\nVerify the result:\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_176.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-onz02o`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_161.png)\n\n## Properties & Relations  (3)Properties of the function, and connections to other functions\nUse [Asymptotic](https:\/\/reference.wolfram.com\/language\/ref\/Asymptotic.html) to compute an asymptotic approximation:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_177.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-d4gzmq`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_162.png)\n\n[LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html) and [InverseLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/InverseLaplaceTransform.html) are mutual inverses:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_178.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-bf3zt1`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_163.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_179.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-4dn9t`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_164.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_180.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-it7bfn`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_165.png)\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_181.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-byd6ra`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_166.png)\n\nUse [NIntegrate](https:\/\/reference.wolfram.com\/language\/ref\/NIntegrate.html) for numerical approximation:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_182.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-f4nw7q`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_167.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_183.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-cganj7`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_168.png)\n\n[NIntegrate](https:\/\/reference.wolfram.com\/language\/ref\/NIntegrate.html) computes the transform for numeric values of the Laplace parameter s:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_184.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-g1xcl`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_169.png)\n\n## Possible Issues  (1)Common pitfalls and unexpected behavior\nSimplification can be required to get back the original form:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_185.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-od1`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_170.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_186.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-fxk`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_171.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_187.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-nrq`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_172.png)\n\n## Neat Examples  (2)Surprising or curious use cases\n[LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html) done in terms of [MeijerG](https:\/\/reference.wolfram.com\/language\/ref\/MeijerG.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_188.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-c92cj`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_173.png)\n\nCreate a table of basic Laplace transforms:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_189.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-mxc5sq`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_190.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-0odb9`\n\nDirect link to example\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_174.png)\n\n# See Also\n[InverseLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/InverseLaplaceTransform.html)  [BilateralLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/BilateralLaplaceTransform.html)  [FourierTransform](https:\/\/reference.wolfram.com\/language\/ref\/FourierTransform.html)  [UnilateralConvolve](https:\/\/reference.wolfram.com\/language\/ref\/UnilateralConvolve.html)  [ZTransform](https:\/\/reference.wolfram.com\/language\/ref\/ZTransform.html)  [Integrate](https:\/\/reference.wolfram.com\/language\/ref\/Integrate.html)  [Piecewise](https:\/\/reference.wolfram.com\/language\/ref\/Piecewise.html)  [TransferFunctionModel](https:\/\/reference.wolfram.com\/language\/ref\/TransferFunctionModel.html)  [MellinTransform](https:\/\/reference.wolfram.com\/language\/ref\/MellinTransform.html)  [Asymptotic](https:\/\/reference.wolfram.com\/language\/ref\/Asymptotic.html)  [FractionalD](https:\/\/reference.wolfram.com\/language\/ref\/FractionalD.html)  [CaputoD](https:\/\/reference.wolfram.com\/language\/ref\/CaputoD.html)\n\n# Tech Notes\n- [Integral Transforms and Related Operations](https:\/\/reference.wolfram.com\/language\/tutorial\/Calculus.html#26017)\n- [Introduction to Fractional Calculus](https:\/\/reference.wolfram.com\/language\/tutorial\/FractionalCalculus.html)\n\n# Related Guides\n- [Fractional Calculus](https:\/\/reference.wolfram.com\/language\/guide\/FractionalCalculus.html)\n- [Fourier Analysis](https:\/\/reference.wolfram.com\/language\/guide\/FourierAnalysis.html)\n- [Generalized Functions](https:\/\/reference.wolfram.com\/language\/guide\/GeneralizedFunctions.html)\n- [Signal Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SignalTransforms.html)\n- [Calculus](https:\/\/reference.wolfram.com\/language\/guide\/Calculus.html)\n- [Integral Transforms](https:\/\/reference.wolfram.com\/language\/guide\/IntegralTransforms.html)\n- [Signal Processing](https:\/\/reference.wolfram.com\/language\/guide\/SignalProcessing.html)\n- [Summation Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SummationTransforms.html)\n\n# Related Links\n- [MathWorld](https:\/\/mathworld.wolfram.com\/LaplaceTransform.html)\n\n# History\nIntroduced in 1999 (4.0)  \\|  [Updated in 2020 (12.2)](https:\/\/reference.wolfram.com\/language\/guide\/SummaryOfNewFeaturesIn122)  ▪  [2023 (13.3)](https:\/\/reference.wolfram.com\/language\/guide\/SummaryOfNewFeaturesIn133)\n\nCite this as:\n\nWolfram Research (1999), LaplaceTransform, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html (updated 2023).\n\nCopy to clipboard.\n\n✖\n\n`Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html (updated 2023).`\n\n![](https:\/\/reference.wolfram.com\/language-assets-v2\/img\/CitationCloseButton.svg)\n\n#### Text\nWolfram Research (1999), LaplaceTransform, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html (updated 2023).\n\nCopy to clipboard.\n\n✖\n\n`Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html (updated 2023).`\n\n#### CMS\nWolfram Language. 1999. \"LaplaceTransform.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html.\n\nCopy to clipboard.\n\n✖\n\n`Wolfram Language. 1999. \"LaplaceTransform.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html.`\n\n#### APA\nWolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html\n\nCopy to clipboard.\n\n✖\n\n`Wolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html`\n\n#### BibTeX\n@misc{reference.wolfram\\_2025\\_laplacetransform, author=\"Wolfram Research\", title=\"{LaplaceTransform}\", year=\"2023\", howpublished=\"\\\\url{https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html}\", note=\\[Accessed: 12-April-2026\\]}\n\nCopy to clipboard.\n\n✖\n\n`@misc{reference.wolfram_2025_laplacetransform, author=\"Wolfram Research\", title=\"{LaplaceTransform}\", year=\"2023\", howpublished=\"\\url{https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html}\", note=[Accessed: 12-April-2026]}`\n\n#### BibLaTeX\n@online{reference.wolfram\\_2025\\_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html}, note=\\[Accessed: 12-April-2026\\]}\n\nCopy to clipboard.\n\n✖\n\n`@online{reference.wolfram_2025_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html}, note=[Accessed: 12-April-2026]}`\n\nGive Feedback\n\n[Top](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html#top)\n\n[Introduction for Programmers](https:\/\/www.wolfram.com\/language\/fast-introduction-for-programmers\/en\/)\n\n[Introductory Book](https:\/\/www.wolfram.com\/language\/elementary-introduction\/)\n\n[Wolfram Function Repository](http:\/\/resources.wolframcloud.com\/FunctionRepository\/)  \\|  [Wolfram Data Repository](https:\/\/datarepository.wolframcloud.com\/)  \\|  [Wolfram Data Drop](https:\/\/datadrop.wolframcloud.com\/)  \\|  [Wolfram Language Products](https:\/\/www.wolfram.com\/products\/)\n\n[Top](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html#top)\n\n- Products\n- [Wolfram\\|One](https:\/\/www.wolfram.com\/wolfram-one\/?source=footer)\n- 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Policy](https:\/\/www.wolfram.com\/legal\/privacy\/wolfram\/?source=footer)  \n[WolframAlpha.com](https:\/\/www.wolframalpha.com\/?source=footer) \\| [WolframCloud.com](https:\/\/www.wolframcloud.com\/?source=footer)\n\n© 2026 [Wolfram](https:\/\/www.wolfram.com\/?source=footer)\n\n© 2026 [Wolfram](https:\/\/www.wolfram.com\/?source=footer) \\| [Legal](https:\/\/www.wolfram.com\/legal\/?source=footer) & [Privacy Policy](https:\/\/www.wolfram.com\/legal\/privacy\/wolfram\/?source=footer) \\|\n\nEnglish\n\n- [English](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html.en?source=footer)\n- [日本語](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html.ja?source=footer)\n- [简体中文](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html.zh?source=footer)\n\n[Find out if you already have access to Wolfram tech through your organization](https:\/\/www.wolfram.com\/siteinfo\/)\n\n×","attrs_readable_markdown":"- [See Also](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n  - [InverseLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/InverseLaplaceTransform.html)\n  - [BilateralLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/BilateralLaplaceTransform.html)\n  - [FourierTransform](https:\/\/reference.wolfram.com\/language\/ref\/FourierTransform.html)\n  - [UnilateralConvolve](https:\/\/reference.wolfram.com\/language\/ref\/UnilateralConvolve.html)\n  - [ZTransform](https:\/\/reference.wolfram.com\/language\/ref\/ZTransform.html)\n  - [Integrate](https:\/\/reference.wolfram.com\/language\/ref\/Integrate.html)\n  - [Piecewise](https:\/\/reference.wolfram.com\/language\/ref\/Piecewise.html)\n  - [TransferFunctionModel](https:\/\/reference.wolfram.com\/language\/ref\/TransferFunctionModel.html)\n  - [MellinTransform](https:\/\/reference.wolfram.com\/language\/ref\/MellinTransform.html)\n  - [Asymptotic](https:\/\/reference.wolfram.com\/language\/ref\/Asymptotic.html)\n  - [FractionalD](https:\/\/reference.wolfram.com\/language\/ref\/FractionalD.html)\n  - [CaputoD](https:\/\/reference.wolfram.com\/language\/ref\/CaputoD.html)\n- [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n  - [Fractional Calculus](https:\/\/reference.wolfram.com\/language\/guide\/FractionalCalculus.html)\n  - [Fourier Analysis](https:\/\/reference.wolfram.com\/language\/guide\/FourierAnalysis.html)\n  - [Generalized Functions](https:\/\/reference.wolfram.com\/language\/guide\/GeneralizedFunctions.html)\n  - [Signal Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SignalTransforms.html)\n  - [Calculus](https:\/\/reference.wolfram.com\/language\/guide\/Calculus.html)\n  - [Integral Transforms](https:\/\/reference.wolfram.com\/language\/guide\/IntegralTransforms.html)\n  - [Signal Processing](https:\/\/reference.wolfram.com\/language\/guide\/SignalProcessing.html)\n  - [Summation Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SummationTransforms.html)\n- [Tech Notes](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n  - [Integral Transforms and Related Operations](https:\/\/reference.wolfram.com\/language\/tutorial\/Calculus.html#26017)\n  - [Introduction to Fractional Calculus](https:\/\/reference.wolfram.com\/language\/tutorial\/FractionalCalculus.html)\n- - [See Also](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n    - [InverseLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/InverseLaplaceTransform.html)\n    - [BilateralLaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/BilateralLaplaceTransform.html)\n    - [FourierTransform](https:\/\/reference.wolfram.com\/language\/ref\/FourierTransform.html)\n    - [UnilateralConvolve](https:\/\/reference.wolfram.com\/language\/ref\/UnilateralConvolve.html)\n    - [ZTransform](https:\/\/reference.wolfram.com\/language\/ref\/ZTransform.html)\n    - [Integrate](https:\/\/reference.wolfram.com\/language\/ref\/Integrate.html)\n    - [Piecewise](https:\/\/reference.wolfram.com\/language\/ref\/Piecewise.html)\n    - [TransferFunctionModel](https:\/\/reference.wolfram.com\/language\/ref\/TransferFunctionModel.html)\n    - [MellinTransform](https:\/\/reference.wolfram.com\/language\/ref\/MellinTransform.html)\n    - [Asymptotic](https:\/\/reference.wolfram.com\/language\/ref\/Asymptotic.html)\n    - [FractionalD](https:\/\/reference.wolfram.com\/language\/ref\/FractionalD.html)\n    - [CaputoD](https:\/\/reference.wolfram.com\/language\/ref\/CaputoD.html)\n  - [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n    - [Fractional Calculus](https:\/\/reference.wolfram.com\/language\/guide\/FractionalCalculus.html)\n    - [Fourier Analysis](https:\/\/reference.wolfram.com\/language\/guide\/FourierAnalysis.html)\n    - [Generalized Functions](https:\/\/reference.wolfram.com\/language\/guide\/GeneralizedFunctions.html)\n    - [Signal Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SignalTransforms.html)\n    - [Calculus](https:\/\/reference.wolfram.com\/language\/guide\/Calculus.html)\n    - [Integral Transforms](https:\/\/reference.wolfram.com\/language\/guide\/IntegralTransforms.html)\n    - [Signal Processing](https:\/\/reference.wolfram.com\/language\/guide\/SignalProcessing.html)\n    - [Summation Transforms](https:\/\/reference.wolfram.com\/language\/guide\/SummationTransforms.html)\n  - [Tech Notes](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\n    - [Integral Transforms and Related Operations](https:\/\/reference.wolfram.com\/language\/tutorial\/Calculus.html#26017)\n    - [Introduction to Fractional Calculus](https:\/\/reference.wolfram.com\/language\/tutorial\/FractionalCalculus.html)\n\nExamples\n\nBasic Examples\n\nScope\n\nBasic Uses\n\nElementary Functions\n\nSpecial Functions\n\nFormal Properties\n\nNumerical Evaluation\n\nFractional Calculus\n\nOptions\n\nAssumptions\n\nGenerateConditions\n\nPrincipal Value\n\nWorking Precision\n\nApplications\n\nOrdinary Differential Equations\n\nFractional Differential Equations\n\nEvaluation of Integrals\n\nOther Applications\n\nProperties & Relations\n\nPossible Issues\n\nNeat Examples\n\n[LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\\[f\\[t\\],t,s\\]\n\nCopy to clipboard.\n\n`LaplaceTransform[f[t],t,s]`\n\ngives the symbolic Laplace transform of f\\[t\\] in the variable t as F\\[s\\] in the variable s.\n\n[LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\\[f\\[t\\],t,![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/1OverscriptBox.png)\\]\n\nCopy to clipboard.\n\n`LaplaceTransform[f[t],t,]`\n\ngives the numeric Laplace transform at the numerical value ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/2OverscriptBox.png).\n\n[LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\\[f\\[t1,…,tn\\],{t1,…,tn},{s1,…,sn}\\]\n\nCopy to clipboard.\n\n`LaplaceTransform[f[t1,…,tn],{t1,…,tn},{s1,…,sn}]`\n\ngives the multidimensional Laplace transform of f\\[t1,…,tn\\].\n\n## Details and Options\n- Laplace transforms are typically used to transform differential and partial differential equations to algebraic equations, solve and then inverse transform back to a solution.\n- Laplace transforms are also extensively used in control theory and signal processing as a way to represent and manipulate linear systems in the form of transfer functions and transfer matrices. The Laplace transform and its inverse are then a way to transform between the time domain and frequency domain.\n- The Laplace transform of a function ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/3.png) is defined to be ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/4.png).\n- The multidimensional Laplace transform is given by ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/5.png).\n- The integral is computed using numerical methods if the third argument, s, is given a numerical value.\n- The asymptotic Laplace transform can be computed using [Asymptotic](https:\/\/reference.wolfram.com\/language\/ref\/Asymptotic.html).\n- The Laplace transform of ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/6.png) exists only for complex values of s in a half-plane ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/7.png).\n- The lower limit of the integral is effectively taken to be ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/9.png), so that the Laplace transform of the Dirac delta function ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/10.png) is equal to 1.  [»](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html#427361964)\n- The following options can be given:\n- |   |   |   |   |\n  |---|---|---|---|\n  |   | [AccuracyGoal](https:\/\/reference.wolfram.com\/language\/ref\/AccuracyGoal.html) | [Automatic](https:\/\/reference.wolfram.com\/language\/ref\/Automatic.html) | digits of absolute accuracy sought |\n  |   | [Assumptions](https:\/\/reference.wolfram.com\/language\/ref\/Assumptions.html) | [\\$Assumptions](https:\/\/reference.wolfram.com\/language\/ref\/$Assumptions.html) | assumptions to make about parameters |\n  |   | [GenerateConditions](https:\/\/reference.wolfram.com\/language\/ref\/GenerateConditions.html) | [False](https:\/\/reference.wolfram.com\/language\/ref\/False.html) | whether to generate answers that involve conditions on parameters |\n  |   | [Method](https:\/\/reference.wolfram.com\/language\/ref\/Method.html) | [Automatic](https:\/\/reference.wolfram.com\/language\/ref\/Automatic.html) | method to use |\n  |   | [PerformanceGoal](https:\/\/reference.wolfram.com\/language\/ref\/PerformanceGoal.html) | [\\$PerformanceGoal](https:\/\/reference.wolfram.com\/language\/ref\/$PerformanceGoal.html) | aspects of performance to optimize |\n  |   | [PrecisionGoal](https:\/\/reference.wolfram.com\/language\/ref\/PrecisionGoal.html) | [Automatic](https:\/\/reference.wolfram.com\/language\/ref\/Automatic.html) | digits of precision sought |\n  |   | [PrincipalValue](https:\/\/reference.wolfram.com\/language\/ref\/PrincipalValue.html) | [False](https:\/\/reference.wolfram.com\/language\/ref\/False.html) | whether to find Cauchy principal value |\n  |   | [WorkingPrecision](https:\/\/reference.wolfram.com\/language\/ref\/WorkingPrecision.html) | [Automatic](https:\/\/reference.wolfram.com\/language\/ref\/Automatic.html) | the precision used in internal computations |\n- Use [GenerateConditions](https:\/\/reference.wolfram.com\/language\/ref\/GenerateConditions.html)\"ConvergenceRegion\" to obtain the region of convergence for the Laplace transform.\n- In [TraditionalForm](https:\/\/reference.wolfram.com\/language\/ref\/TraditionalForm.html), [LaplaceTransform](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html) is output using ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/11.png). [»](https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html#2079194920)\n\n## Examples\nopen all close all\n\n## Basic Examples  (4)Summary of the most common use cases\n## Scope  (67)Survey of the scope of standard use cases\n### Basic Uses  (4)\n### Elementary Functions  (13)\n### Special Functions  (10)\n### Piecewise Functions  (9)\n### Periodic Functions  (5)\n### Generalized Functions  (5)\n### Multivariate Functions  (9)\nBivariate Laplace transform of a constant:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_83.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-l51olx`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_75.png)\n\nExponential function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_84.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-iftr0v`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_76.png)\n\nPower function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_85.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-xn08dr`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_77.png)\n\n[BesselJ](https:\/\/reference.wolfram.com\/language\/ref\/BesselJ.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_86.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-pleo5d`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_78.png)\n\nSquare root:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_87.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-wilqmz`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_79.png)\n\nComposition of cosine and square root:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_88.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-m19040`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_80.png)\n\nLaplace transform of a multivariate power function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_89.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-jvc1kh`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_81.png)\n\nCosine:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_90.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-w44hxj`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_82.png)\n\nLogarithm:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_91.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-nse2y4`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_83.png)\n### Formal Properties  (6)\n### Numerical Evaluation  (3)\n### Fractional Calculus  (3)\n\n## Options  (4)Common values & functionality for each option\n### Assumptions  (1)\nSpecify the range for a parameter using [Assumptions](https:\/\/reference.wolfram.com\/language\/ref\/Assumptions.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_118.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-fkqovc`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_108.png)\n\n### GenerateConditions  (1)\nUse [GenerateConditions](https:\/\/reference.wolfram.com\/language\/ref\/GenerateConditions.html)\\-\\>[True](https:\/\/reference.wolfram.com\/language\/ref\/True.html) to get parameter conditions for when a result is valid:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_119.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-6pnei`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_109.png)\n### Principal Value  (1)\n### Working Precision  (1)\n\n## Applications  (12)Sample problems that can be solved with this function\n### Ordinary Differential Equations  (5)\n### Fractional Differential Equations  (3)\nSolve a fractional-order differential equation using Laplace transforms:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_142.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-5yhytr`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_131.png)\n\nSolve for the Laplace transform:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_143.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-8g7evs`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_132.png)\n\nFind the inverse transform:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_144.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-203z1c`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_133.png)\n\nPlot the solution:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_145.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-oepin2`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_134.png)\n\nFind the solution directly using [DSolve](https:\/\/reference.wolfram.com\/language\/ref\/DSolve.html):\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_146.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-iiovhh`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_135.png)\n\nSolve the following fractional integro-differential equation:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_147.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-sboa2x`\n\nDirect link to example\n\nSolve for the Laplace transform:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_148.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-6hxn7n`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_136.png)\n\nFind the inverse transform:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_149.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-ik5ypj`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_137.png)\n\nFind the solution directly using [DSolve](https:\/\/reference.wolfram.com\/language\/ref\/DSolve.html):\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_150.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-hvse9n`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_138.png)\n\nThe following equation describes a fractional harmonic oscillator of order 1.9:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_151.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-usm5e4`\n\nDirect link to example\n\nSolve for the Laplace transform:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_152.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-zsldvm`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_139.png)\n\nFind the inverse transform:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_153.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-x8hdfh`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_140.png)\n\nPlot the solution:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_154.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-msydxu`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_141.png)\n\nFind the solution directly using [DSolve](https:\/\/reference.wolfram.com\/language\/ref\/DSolve.html):\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/I_155.png)\n\n`https:\/\/wolfram.com\/xid\/0h2pdw27n2-ogzgeo`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/LaplaceTransform.en\/O_142.png)\n### Evaluation of Integrals  (2)\n### Other Applications  (2)\n## Properties & Relations  (3)Properties of the function, and connections to other functions\n## Possible Issues  (1)Common pitfalls and unexpected behavior\n## Neat Examples  (2)Surprising or curious use cases\n\nWolfram Research (1999), LaplaceTransform, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html (updated 2023).\n\nCopy to clipboard.\n\n`Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html (updated 2023).`\n\n#### Text\nWolfram Research (1999), LaplaceTransform, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html (updated 2023).\n\nCopy to clipboard.\n\n`Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html (updated 2023).`\n\n#### CMS\nWolfram Language. 1999. \"LaplaceTransform.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html.\n\nCopy to clipboard.\n\n`Wolfram Language. 1999. \"LaplaceTransform.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html.`\n\n#### APA\nWolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html\n\nCopy to clipboard.\n\n`Wolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html`\n\n#### BibTeX\n@misc{reference.wolfram\\_2025\\_laplacetransform, author=\"Wolfram Research\", title=\"{LaplaceTransform}\", year=\"2023\", howpublished=\"\\\\url{https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html}\", note=\\[Accessed: 12-April-2026\\]}\n\nCopy to clipboard.\n\n`@misc{reference.wolfram_2025_laplacetransform, author=\"Wolfram Research\", title=\"{LaplaceTransform}\", year=\"2023\", howpublished=\"\\url{https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html}\", note=[Accessed: 12-April-2026]}`\n\n#### BibLaTeX\n@online{reference.wolfram\\_2025\\_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html}, note=\\[Accessed: 12-April-2026\\]}\n\nCopy to clipboard.\n\n`@online{reference.wolfram_2025_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html}, note=[Accessed: 12-April-2026]}`","meta_canonical":null}

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Meta Title	LaplaceTransform—Wolfram Documentation
Meta Description	LaplaceTransform[f[t], t, s] gives the symbolic Laplace transform of f[t] in the variable t as F[s] in the variable s. LaplaceTransform[f[t], t, OverscriptBox[s, ^]] gives the numeric Laplace transform at the numerical value OverscriptBox[s, ^]. LaplaceTransform[f[t1, ..., tn], {t1, ..., tn}, {s1, ..., sn}] gives the multidimensional Laplace transform of f[t1, ..., tn].
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Boilerpipe Text	See Also InverseLaplaceTransform BilateralLaplaceTransform FourierTransform UnilateralConvolve ZTransform Integrate Piecewise TransferFunctionModel MellinTransform Asymptotic FractionalD CaputoD Related Guides Fractional Calculus Fourier Analysis Generalized Functions Signal Transforms Calculus Integral Transforms Signal Processing Summation Transforms Tech Notes Integral Transforms and Related Operations Introduction to Fractional Calculus See Also InverseLaplaceTransform BilateralLaplaceTransform FourierTransform UnilateralConvolve ZTransform Integrate Piecewise TransferFunctionModel MellinTransform Asymptotic FractionalD CaputoD Related Guides Fractional Calculus Fourier Analysis Generalized Functions Signal Transforms Calculus Integral Transforms Signal Processing Summation Transforms Tech Notes Integral Transforms and Related Operations Introduction to Fractional Calculus Examples Basic Examples Scope Basic Uses Elementary Functions Special Functions Formal Properties Numerical Evaluation Fractional Calculus Options Assumptions GenerateConditions Principal Value Working Precision Applications Ordinary Differential Equations Fractional Differential Equations Evaluation of Integrals Other Applications Properties & Relations Possible Issues Neat Examples LaplaceTransform [ f [ t ] , t , s ] Copy to clipboard. LaplaceTransform[f[t],t,s] gives the symbolic Laplace transform of f [ t ] in the variable t as F [ s ] in the variable s . LaplaceTransform [ f [ t ] , t , ] Copy to clipboard. LaplaceTransform[f[t],t,] gives the numeric Laplace transform at the numerical value . LaplaceTransform [ f [ t 1 , … , t n ] , { t 1 , … , t n } , { s 1 , … , s n } ] Copy to clipboard. LaplaceTransform[f[t1,…,tn],{t1,…,tn},{s1,…,sn}] gives the multidimensional Laplace transform of f [ t 1 , … , t n ] . Details and Options Laplace transforms are typically used to transform differential and partial differential equations to algebraic equations, solve and then inverse transform back to a solution. Laplace transforms are also extensively used in control theory and signal processing as a way to represent and manipulate linear systems in the form of transfer functions and transfer matrices. The Laplace transform and its inverse are then a way to transform between the time domain and frequency domain. The Laplace transform of a function is defined to be . The multidimensional Laplace transform is given by . The integral is computed using numerical methods if the third argument, s , is given a numerical value. The asymptotic Laplace transform can be computed using Asymptotic . The Laplace transform of exists only for complex values of s in a half-plane . The lower limit of the integral is effectively taken to be , so that the Laplace transform of the Dirac delta function is equal to 1. » The following options can be given: AccuracyGoal Automatic digits of absolute accuracy sought Assumptions $Assumptions assumptions to make about parameters GenerateConditions False whether to generate answers that involve conditions on parameters Method Automatic method to use PerformanceGoal $PerformanceGoal aspects of performance to optimize PrecisionGoal Automatic digits of precision sought PrincipalValue False whether to find Cauchy principal value WorkingPrecision Automatic the precision used in internal computations Use GenerateConditions  "ConvergenceRegion" to obtain the region of convergence for the Laplace transform. In TraditionalForm , LaplaceTransform is output using . » Examples open all close all Basic Examples (4) Summary of the most common use cases Scope (67) Survey of the scope of standard use cases Basic Uses (4) Elementary Functions (13) Special Functions (10) Piecewise Functions (9) Periodic Functions (5) Generalized Functions (5) Multivariate Functions (9) Bivariate Laplace transform of a constant: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-l51olx Direct link to example Out[1]= 1 Exponential function: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-iftr0v Direct link to example Out[1]= 1 Power function: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-xn08dr Direct link to example Out[1]= 1 BesselJ : Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-pleo5d Direct link to example Out[1]= 1 Square root: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-wilqmz Direct link to example Out[1]= 1 Composition of cosine and square root: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-m19040 Direct link to example Out[1]= 1 Laplace transform of a multivariate power function: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-jvc1kh Direct link to example Out[1]= 1 Cosine: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-w44hxj Direct link to example Out[1]= 1 Logarithm: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-nse2y4 Direct link to example Out[1]= 1 Formal Properties (6) Numerical Evaluation (3) Fractional Calculus (3) Options (4) Common values & functionality for each option Assumptions (1) Specify the range for a parameter using Assumptions : Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-fkqovc Direct link to example Out[1]= 1 GenerateConditions (1) Use GenerateConditions -> True to get parameter conditions for when a result is valid: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-6pnei Direct link to example Out[1]= 1 Principal Value (1) Working Precision (1) Applications (12) Sample problems that can be solved with this function Ordinary Differential Equations (5) Fractional Differential Equations (3) Solve a fractional-order differential equation using Laplace transforms: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-5yhytr Direct link to example Out[1]= 1 Solve for the Laplace transform: Copy to clipboard. In[2]:= 2 https://wolfram.com/xid/0h2pdw27n2-8g7evs Direct link to example Out[2]= 2 Find the inverse transform: Copy to clipboard. In[3]:= 3 https://wolfram.com/xid/0h2pdw27n2-203z1c Direct link to example Out[3]= 3 Plot the solution: Copy to clipboard. In[4]:= 4 https://wolfram.com/xid/0h2pdw27n2-oepin2 Direct link to example Out[4]= 4 Find the solution directly using DSolve : Copy to clipboard. In[5]:= 5 https://wolfram.com/xid/0h2pdw27n2-iiovhh Direct link to example Out[5]= 5 Solve the following fractional integro-differential equation: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-sboa2x Direct link to example Solve for the Laplace transform: Copy to clipboard. In[2]:= 2 https://wolfram.com/xid/0h2pdw27n2-6hxn7n Direct link to example Out[2]= 2 Find the inverse transform: Copy to clipboard. In[3]:= 3 https://wolfram.com/xid/0h2pdw27n2-ik5ypj Direct link to example Out[3]= 3 Find the solution directly using DSolve : Copy to clipboard. In[4]:= 4 https://wolfram.com/xid/0h2pdw27n2-hvse9n Direct link to example Out[4]= 4 The following equation describes a fractional harmonic oscillator of order 1.9: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0h2pdw27n2-usm5e4 Direct link to example Solve for the Laplace transform: Copy to clipboard. In[2]:= 2 https://wolfram.com/xid/0h2pdw27n2-zsldvm Direct link to example Out[2]= 2 Find the inverse transform: Copy to clipboard. In[3]:= 3 https://wolfram.com/xid/0h2pdw27n2-x8hdfh Direct link to example Out[3]= 3 Plot the solution: Copy to clipboard. In[4]:= 4 https://wolfram.com/xid/0h2pdw27n2-msydxu Direct link to example Out[4]= 4 Find the solution directly using DSolve : Copy to clipboard. In[5]:= 5 https://wolfram.com/xid/0h2pdw27n2-ogzgeo Direct link to example Out[5]= 5 Evaluation of Integrals (2) Other Applications (2) Properties & Relations (3) Properties of the function, and connections to other functions Possible Issues (1) Common pitfalls and unexpected behavior Neat Examples (2) Surprising or curious use cases Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023). Copy to clipboard. Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023). Text Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023). Copy to clipboard. Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023). CMS Wolfram Language. 1999. "LaplaceTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/LaplaceTransform.html. Copy to clipboard. Wolfram Language. 1999. "LaplaceTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/LaplaceTransform.html. APA Wolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LaplaceTransform.html Copy to clipboard. Wolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LaplaceTransform.html BibTeX @misc{reference.wolfram_2025_laplacetransform, author="Wolfram Research", title="{LaplaceTransform}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/LaplaceTransform.html}", note=[Accessed: 12-April-2026]} Copy to clipboard. @misc{reference.wolfram_2025_laplacetransform, author="Wolfram Research", title="{LaplaceTransform}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/LaplaceTransform.html}", note=[Accessed: 12-April-2026]} BibLaTeX @online{reference.wolfram_2025_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https://reference.wolfram.com/language/ref/LaplaceTransform.html}, note=[Accessed: 12-April-2026]} Copy to clipboard. @online{reference.wolfram_2025_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https://reference.wolfram.com/language/ref/LaplaceTransform.html}, note=[Accessed: 12-April-2026]}
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[Piecewise](https://reference.wolfram.com/language/ref/Piecewise.html) - [TransferFunctionModel](https://reference.wolfram.com/language/ref/TransferFunctionModel.html) - [MellinTransform](https://reference.wolfram.com/language/ref/MellinTransform.html) - [Asymptotic](https://reference.wolfram.com/language/ref/Asymptotic.html) - [FractionalD](https://reference.wolfram.com/language/ref/FractionalD.html) - [CaputoD](https://reference.wolfram.com/language/ref/CaputoD.html) - [Related Guides](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [Fractional Calculus](https://reference.wolfram.com/language/guide/FractionalCalculus.html) - [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.html) - [Generalized Functions](https://reference.wolfram.com/language/guide/GeneralizedFunctions.html) - [Signal Transforms](https://reference.wolfram.com/language/guide/SignalTransforms.html) - [Calculus](https://reference.wolfram.com/language/guide/Calculus.html) - [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.html) - [Signal Processing](https://reference.wolfram.com/language/guide/SignalProcessing.html) - [Summation Transforms](https://reference.wolfram.com/language/guide/SummationTransforms.html) - [Tech Notes](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [Integral Transforms and Related Operations](https://reference.wolfram.com/language/tutorial/Calculus.html#26017) - [Introduction to Fractional Calculus](https://reference.wolfram.com/language/tutorial/FractionalCalculus.html) - - [See Also](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [InverseLaplaceTransform](https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html) - [BilateralLaplaceTransform](https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.html) - [FourierTransform](https://reference.wolfram.com/language/ref/FourierTransform.html) - [UnilateralConvolve](https://reference.wolfram.com/language/ref/UnilateralConvolve.html) - [ZTransform](https://reference.wolfram.com/language/ref/ZTransform.html) - [Integrate](https://reference.wolfram.com/language/ref/Integrate.html) - [Piecewise](https://reference.wolfram.com/language/ref/Piecewise.html) - [TransferFunctionModel](https://reference.wolfram.com/language/ref/TransferFunctionModel.html) - [MellinTransform](https://reference.wolfram.com/language/ref/MellinTransform.html) - [Asymptotic](https://reference.wolfram.com/language/ref/Asymptotic.html) - [FractionalD](https://reference.wolfram.com/language/ref/FractionalD.html) - [CaputoD](https://reference.wolfram.com/language/ref/CaputoD.html) - [Related Guides](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [Fractional Calculus](https://reference.wolfram.com/language/guide/FractionalCalculus.html) - [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.html) - [Generalized Functions](https://reference.wolfram.com/language/guide/GeneralizedFunctions.html) - [Signal Transforms](https://reference.wolfram.com/language/guide/SignalTransforms.html) - [Calculus](https://reference.wolfram.com/language/guide/Calculus.html) - [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.html) - [Signal Processing](https://reference.wolfram.com/language/guide/SignalProcessing.html) - [Summation Transforms](https://reference.wolfram.com/language/guide/SummationTransforms.html) - [Tech Notes](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [Integral Transforms and Related Operations](https://reference.wolfram.com/language/tutorial/Calculus.html#26017) - [Introduction to Fractional Calculus](https://reference.wolfram.com/language/tutorial/FractionalCalculus.html) [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html)\[f\[t\],t,s\] gives the symbolic Laplace transform of f\[t\] in the variable t as F\[s\] in the variable s. [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html)\[f\[t\],t,![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/28OverscriptBox.png)\] gives the numeric Laplace transform at the numerical value ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/29OverscriptBox.png). [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html)\[f\[t1,…,tn\],{t1,…,tn},{s1,…,sn}\] gives the multidimensional Laplace transform of f\[t1,…,tn\]. Details and Options ![Details and Options](https://reference.wolfram.com/language-assets-v2/img/DetailsOptions-gray.png) ![Details and Options](https://reference.wolfram.com/language-assets-v2/img/DetailsOptions-hover.png) Examples Basic Examples Scope Basic Uses Elementary Functions Special Functions ![Show More](https://reference.wolfram.com/language-assets-v2/img/Options-hamburger-gray.svg) ![Show More](https://reference.wolfram.com/language-assets-v2/img/Options-hamburger-hover.svg) Piecewise Functions Periodic Functions Generalized Functions Multivariate Functions Formal Properties Numerical Evaluation Fractional Calculus Options Assumptions GenerateConditions Principal Value Working Precision Applications Ordinary Differential Equations Fractional Differential Equations Evaluation of Integrals Other Applications Properties & Relations Possible Issues Neat Examples See Also Tech Notes Related Guides Related Links History Cite this Page BUILT-IN SYMBOL - [See Also](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [InverseLaplaceTransform](https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html) - [BilateralLaplaceTransform](https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.html) - [FourierTransform](https://reference.wolfram.com/language/ref/FourierTransform.html) - [UnilateralConvolve](https://reference.wolfram.com/language/ref/UnilateralConvolve.html) - [ZTransform](https://reference.wolfram.com/language/ref/ZTransform.html) - [Integrate](https://reference.wolfram.com/language/ref/Integrate.html) - [Piecewise](https://reference.wolfram.com/language/ref/Piecewise.html) - [TransferFunctionModel](https://reference.wolfram.com/language/ref/TransferFunctionModel.html) - [MellinTransform](https://reference.wolfram.com/language/ref/MellinTransform.html) - [Asymptotic](https://reference.wolfram.com/language/ref/Asymptotic.html) - [FractionalD](https://reference.wolfram.com/language/ref/FractionalD.html) - [CaputoD](https://reference.wolfram.com/language/ref/CaputoD.html) - [Related Guides](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [Fractional Calculus](https://reference.wolfram.com/language/guide/FractionalCalculus.html) - [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.html) - [Generalized Functions](https://reference.wolfram.com/language/guide/GeneralizedFunctions.html) - [Signal Transforms](https://reference.wolfram.com/language/guide/SignalTransforms.html) - [Calculus](https://reference.wolfram.com/language/guide/Calculus.html) - [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.html) - [Signal Processing](https://reference.wolfram.com/language/guide/SignalProcessing.html) - [Summation Transforms](https://reference.wolfram.com/language/guide/SummationTransforms.html) - [Tech Notes](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [Integral Transforms and Related Operations](https://reference.wolfram.com/language/tutorial/Calculus.html#26017) - [Introduction to Fractional Calculus](https://reference.wolfram.com/language/tutorial/FractionalCalculus.html) - - [See Also](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [InverseLaplaceTransform](https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html) - [BilateralLaplaceTransform](https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.html) - [FourierTransform](https://reference.wolfram.com/language/ref/FourierTransform.html) - [UnilateralConvolve](https://reference.wolfram.com/language/ref/UnilateralConvolve.html) - [ZTransform](https://reference.wolfram.com/language/ref/ZTransform.html) - [Integrate](https://reference.wolfram.com/language/ref/Integrate.html) - [Piecewise](https://reference.wolfram.com/language/ref/Piecewise.html) - [TransferFunctionModel](https://reference.wolfram.com/language/ref/TransferFunctionModel.html) - [MellinTransform](https://reference.wolfram.com/language/ref/MellinTransform.html) - [Asymptotic](https://reference.wolfram.com/language/ref/Asymptotic.html) - [FractionalD](https://reference.wolfram.com/language/ref/FractionalD.html) - [CaputoD](https://reference.wolfram.com/language/ref/CaputoD.html) - [Related Guides](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [Fractional Calculus](https://reference.wolfram.com/language/guide/FractionalCalculus.html) - [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.html) - [Generalized Functions](https://reference.wolfram.com/language/guide/GeneralizedFunctions.html) - [Signal Transforms](https://reference.wolfram.com/language/guide/SignalTransforms.html) - [Calculus](https://reference.wolfram.com/language/guide/Calculus.html) - [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.html) - [Signal Processing](https://reference.wolfram.com/language/guide/SignalProcessing.html) - [Summation Transforms](https://reference.wolfram.com/language/guide/SummationTransforms.html) - [Tech Notes](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [Integral Transforms and Related Operations](https://reference.wolfram.com/language/tutorial/Calculus.html#26017) - [Introduction to Fractional Calculus](https://reference.wolfram.com/language/tutorial/FractionalCalculus.html) # LaplaceTransformCopy to clipboard. ✖ `LaplaceTransform` [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html)\[f\[t\],t,s\] Copy to clipboard. ✖ `LaplaceTransform[f[t],t,s]` gives the symbolic Laplace transform of f\[t\] in the variable t as F\[s\] in the variable s. [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html)\[f\[t\],t,![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/1OverscriptBox.png)\] Copy to clipboard. ✖ `LaplaceTransform[f[t],t,]` gives the numeric Laplace transform at the numerical value ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/2OverscriptBox.png). [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html)\[f\[t1,…,tn\],{t1,…,tn},{s1,…,sn}\] Copy to clipboard. ✖ `LaplaceTransform[f[t1,…,tn],{t1,…,tn},{s1,…,sn}]` gives the multidimensional Laplace transform of f\[t1,…,tn\]. # Details and Options ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/details_1.png) ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/details_2.png) - Laplace transforms are typically used to transform differential and partial differential equations to algebraic equations, solve and then inverse transform back to a solution. - Laplace transforms are also extensively used in control theory and signal processing as a way to represent and manipulate linear systems in the form of transfer functions and transfer matrices. The Laplace transform and its inverse are then a way to transform between the time domain and frequency domain. - The Laplace transform of a function ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/3.png) is defined to be ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/4.png). - The multidimensional Laplace transform is given by ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/5.png). - The integral is computed using numerical methods if the third argument, s, is given a numerical value. - The asymptotic Laplace transform can be computed using [Asymptotic](https://reference.wolfram.com/language/ref/Asymptotic.html). - The Laplace transform of ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/6.png) exists only for complex values of s in a half-plane ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/7.png). - The lower limit of the integral is effectively taken to be ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/9.png), so that the Laplace transform of the Dirac delta function ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/10.png) is equal to 1. [»](https://reference.wolfram.com/language/ref/LaplaceTransform.html#427361964) - The following options can be given: - \| \| \| \| \| \|---\|---\|---\|---\| \| \| [AccuracyGoal](https://reference.wolfram.com/language/ref/AccuracyGoal.html) \| [Automatic](https://reference.wolfram.com/language/ref/Automatic.html) \| digits of absolute accuracy sought \| \| \| [Assumptions](https://reference.wolfram.com/language/ref/Assumptions.html) \| [\$Assumptions](https://reference.wolfram.com/language/ref/$Assumptions.html) \| assumptions to make about parameters \| \| \| [GenerateConditions](https://reference.wolfram.com/language/ref/GenerateConditions.html) \| [False](https://reference.wolfram.com/language/ref/False.html) \| whether to generate answers that involve conditions on parameters \| \| \| [Method](https://reference.wolfram.com/language/ref/Method.html) \| [Automatic](https://reference.wolfram.com/language/ref/Automatic.html) \| method to use \| \| \| [PerformanceGoal](https://reference.wolfram.com/language/ref/PerformanceGoal.html) \| [\$PerformanceGoal](https://reference.wolfram.com/language/ref/$PerformanceGoal.html) \| aspects of performance to optimize \| \| \| [PrecisionGoal](https://reference.wolfram.com/language/ref/PrecisionGoal.html) \| [Automatic](https://reference.wolfram.com/language/ref/Automatic.html) \| digits of precision sought \| \| \| [PrincipalValue](https://reference.wolfram.com/language/ref/PrincipalValue.html) \| [False](https://reference.wolfram.com/language/ref/False.html) \| whether to find Cauchy principal value \| \| \| [WorkingPrecision](https://reference.wolfram.com/language/ref/WorkingPrecision.html) \| [Automatic](https://reference.wolfram.com/language/ref/Automatic.html) \| the precision used in internal computations \| - Use [GenerateConditions](https://reference.wolfram.com/language/ref/GenerateConditions.html)"ConvergenceRegion" to obtain the region of convergence for the Laplace transform. - In [TraditionalForm](https://reference.wolfram.com/language/ref/TraditionalForm.html), [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html) is output using ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/11.png). [»](https://reference.wolfram.com/language/ref/LaplaceTransform.html#2079194920) # Examples open all close all ## Basic Examples (4)Summary of the most common use cases Compute the Laplace transform of a function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_1.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-52saw3` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_1.png) Define a piecewise function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_2.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-fl1sjt` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_3.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-60cdu` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_2.png) Compute its Laplace transform: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_4.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-7hhqvo` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_3.png) Compute the transform at a single point: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_5.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-fqa72t` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_4.png) Compute the Laplace transform of a multivariate function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_6.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-0ewuhr` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_5.png) Define a multivariate piecewise function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_7.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-fom62v` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_8.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-69gxmn` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_6.png) Compute its Laplace transform: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_9.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-eqq3xl` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_7.png) ## Scope (67)Survey of the scope of standard use cases ### Basic Uses (4) Laplace transform of a function for a symbolic parameter s: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_10.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-vlts3p` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_8.png) Laplace transforms of trigonometric functions: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_11.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-qf9a9` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_9.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_12.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-bargvx` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_10.png) Evaluate the Laplace transform for a numerical value of the parameter s: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_13.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-4ycwlj` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_11.png) [TraditionalForm](https://reference.wolfram.com/language/ref/TraditionalForm.html) formatting: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_14.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-tpw3x2` Direct link to example ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_12.png) ### Elementary Functions (13) Laplace transform of a power function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_15.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-bxq0e5` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_13.png) Square root function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_16.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-l3vqf0` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_14.png) Laplace transforms of polynomials: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_17.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-hevqei` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_15.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_18.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-d05omh` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_16.png) Exponential function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_19.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-kvdxn3` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_17.png) Product of an exponential and a linear function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_20.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-104rjl` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_18.png) Expressions involving trigonometric functions: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_21.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-tqq1h2` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_19.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_22.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-gd0kyr` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_20.png) Expressions involving hyperbolic functions: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_23.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-i8tzxa` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_21.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_24.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-3t5n0r` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_22.png) Ratio of an exponential and a linear function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_25.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-gbulo8` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_23.png) Ratio of sine and linear functions: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_26.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-bsm8py` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_24.png) Composition of elementary functions: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_27.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-s9a3k` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_25.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_28.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-lwqt7u` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_26.png) Logarithmic function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_29.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-5bwuuf` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_27.png) Product of logarithmic and power functions: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_30.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-p2nd7m` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_28.png) Square of a logarithmic function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_31.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-j4w84x` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_29.png) ### Special Functions (10) Laplace transform of error and square root functions composition: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_32.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-k0nw19` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_30.png) Bessel functions: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_33.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-9irs5p` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_31.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_34.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-2gxc7` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_32.png) Products involving Bessel functions: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_35.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-pbgdhr` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_33.png) Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_36.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-c04r43` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_34.png) Sine integral function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_37.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-q8ehy8` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_35.png) Laguerre polynomials: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_38.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-xzqhic` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_36.png) Airy function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_39.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-hso455` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_37.png) Chebyshev polynomial: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_40.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-rt1kwi` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_38.png) Struve function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_41.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-p1ki8t` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_39.png) Fresnel function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_42.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-5bn4go` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_40.png) Gamma function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_43.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-2yz0oy` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_41.png) Hypergeometric function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_44.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-4u6kzh` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_42.png) ### Piecewise Functions (9) Laplace transform of a piecewise function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_45.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-h4lxo8` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_46.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-b05sk8` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_43.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_47.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-ruge2u` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_44.png) Restriction of a sine function to a half-period: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_48.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-2hml80` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_49.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-iwlcqb` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_45.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_50.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-g1bigg` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_46.png) Exponential function with a left cutoff: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_51.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-mw0vs1` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_52.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-iyy5ib` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_47.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_53.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-hj9cvx` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_48.png) Triangular function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_54.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-kor8nt` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_55.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-09ydxm` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_49.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_56.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-ldq8my` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_50.png) Polynomial function with a left cutoff: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_57.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-499t0o` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_58.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-e39213` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_51.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_59.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-7yy7ev` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_52.png) Ramp: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_60.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-ydoxba` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_53.png) [UnitStep](https://reference.wolfram.com/language/ref/UnitStep.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_61.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-sqqyn7` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_54.png) Product of [UnitStep](https://reference.wolfram.com/language/ref/UnitStep.html) and cosine functions: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_62.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-48rqn1` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_55.png) Laplace transform of [Floor](https://reference.wolfram.com/language/ref/Floor.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_63.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-6akvqj` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_56.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_64.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-bmflvc` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_57.png) ### Periodic Functions (5) Laplace transform of [SquareWave](https://reference.wolfram.com/language/ref/SquareWave.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_65.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-7ktv9h` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_58.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_66.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-ycjry8` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_59.png) [TriangleWave](https://reference.wolfram.com/language/ref/TriangleWave.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_67.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-4qy61o` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_60.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_68.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-2avn8l` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_61.png) [SawtoothWave](https://reference.wolfram.com/language/ref/SawtoothWave.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_69.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-u745mq` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_62.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_70.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-emltg7` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_63.png) Full-wave-rectified function with period ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/12.png): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_71.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-nb3raf` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_64.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_72.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-byq9w3` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_65.png) Rectified wave: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_73.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-no882i` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_74.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-fig26d` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_66.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_75.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-zqhvt8` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_67.png) ### Generalized Functions (5) Laplace transform of [HeavisideTheta](https://reference.wolfram.com/language/ref/HeavisideTheta.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_76.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-pg20cq` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_68.png) [DiracDelta](https://reference.wolfram.com/language/ref/DiracDelta.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_77.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-yvqfg2` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_69.png) Derivative of [DiracDelta](https://reference.wolfram.com/language/ref/DiracDelta.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_78.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-f8njof` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_70.png) [HeavisideLambda](https://reference.wolfram.com/language/ref/HeavisideLambda.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_79.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-uw6trd` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_71.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_80.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-1faykd` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_72.png) [HeavisidePi](https://reference.wolfram.com/language/ref/HeavisidePi.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_81.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-uv2j6r` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_73.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_82.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-2m3had` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_74.png) ### Multivariate Functions (9) Bivariate Laplace transform of a constant: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_83.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-l51olx` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_75.png) Exponential function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_84.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-iftr0v` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_76.png) Power function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_85.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-xn08dr` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_77.png) [BesselJ](https://reference.wolfram.com/language/ref/BesselJ.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_86.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-pleo5d` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_78.png) Square root: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_87.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-wilqmz` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_79.png) Composition of cosine and square root: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_88.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-m19040` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_80.png) Laplace transform of a multivariate power function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_89.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-jvc1kh` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_81.png) Cosine: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_90.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-w44hxj` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_82.png) Logarithm: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_91.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-nse2y4` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_83.png) ### Formal Properties (6) The Laplace transform is a linear operator: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_92.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-sv38rg` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_84.png) Laplace transform of ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/13.png) is the Laplace transform of ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/14.png) evaluated at ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/15.png): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_93.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-3p91yz` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_85.png) Laplace transform of a first-order derivative: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_94.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-s0daap` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_86.png) Laplace transform of a second-order derivative: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_95.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-uwnfg0` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_87.png) Laplace transform of a product with monomials: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_96.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-kfzvq4` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_88.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_97.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-btnn48` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_89.png) Laplace transform threads itself over equations: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_98.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-kyrnxk` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_90.png) ### Numerical Evaluation (3) Calculate the Laplace transform at a single point: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_99.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-fhhyve` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_91.png) Alternatively, calculate the Laplace transform symbolically: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_100.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-1ten3m` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_92.png) Then evaluate it for specific value of ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/16.png): Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_101.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-bmvmnm` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_93.png) Plot the Laplace transform using numerical values only: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_102.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-lyv72q` Direct link to example Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_103.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-ir3wks` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_94.png) For some functions, the Laplace transform cannot be evaluated symbolically: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_104.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-gcfdhk` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_95.png) Evaluate the Laplace transform numerically and plot it: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_105.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-bomtml` Direct link to example Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_106.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-cs3x94` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_96.png) Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_107.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-nd2svs` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_97.png) Calculate a multivariate Laplace transform at a single point in the plane: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_108.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-wqufw9` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_98.png) ### Fractional Calculus (3) Laplace transform of the [MittagLefflerE](https://reference.wolfram.com/language/ref/MittagLefflerE.html) functions: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_109.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-9fi54c` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_99.png) [ComplexPlot](https://reference.wolfram.com/language/ref/ComplexPlot.html) in the ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/17.png)\-domain: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_110.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-3nzaga` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_100.png) Inverse Laplace transform to the time domain: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_111.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-ox1krh` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_101.png) Laplace transform of the [MittagLefflerE](https://reference.wolfram.com/language/ref/MittagLefflerE.html) functions involving parameters: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_112.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-s6e1zo` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_102.png) Inverse Laplace transform to the time domain: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_113.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-2xte5w` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_103.png) Laplace transform of the [CaputoD](https://reference.wolfram.com/language/ref/CaputoD.html) fractional derivative: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_114.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-dyaoq6` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_104.png) Apply to sine function: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_115.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-dvqm5x` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_105.png) Compare this with the [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html) of the [CaputoD](https://reference.wolfram.com/language/ref/CaputoD.html) derivative of the sine function: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_116.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-p7qk4m` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_106.png) Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_117.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-csb1rs` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_107.png) ## Options (4)Common values & functionality for each option ### Assumptions (1) Specify the range for a parameter using [Assumptions](https://reference.wolfram.com/language/ref/Assumptions.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_118.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-fkqovc` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_108.png) ### GenerateConditions (1) Use [GenerateConditions](https://reference.wolfram.com/language/ref/GenerateConditions.html)\-\>[True](https://reference.wolfram.com/language/ref/True.html) to get parameter conditions for when a result is valid: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_119.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-6pnei` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_109.png) ### Principal Value (1) The Laplace transform of the following function is not defined due to the singularity at ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/18.png): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_120.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-byox9b` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_110.png) Use [PrincipalValue](https://reference.wolfram.com/language/ref/PrincipalValue.html) to obtain the Cauchy principal value for the integral: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_121.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-ly80t6` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_111.png) ### Working Precision (1) Use [WorkingPrecision](https://reference.wolfram.com/language/ref/WorkingPrecision.html) to obtain a result with arbitrary precision: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_122.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-cm1yhe` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_112.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_123.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-jp0k2` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_113.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_124.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-gylrkk` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_114.png) ## Applications (12)Sample problems that can be solved with this function ### Ordinary Differential Equations (5) Solve a differential equation using Laplace transforms: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_125.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-z3evmw` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_115.png) Solve for the Laplace transform: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_126.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-spc3fq` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_116.png) Find the inverse transform: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_127.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-8kjf6g` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_117.png) Plot the solution: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_128.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-kwb15a` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_118.png) Find the solution directly using [DSolve](https://reference.wolfram.com/language/ref/DSolve.html): Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_129.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-w27252` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_119.png) Solve the following differential equation: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_130.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-q5z19` Direct link to example Solve for the Laplace transform: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_131.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-suvmxo` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_120.png) Find the inverse transform: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_132.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-jycu84` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_121.png) Plot the solution: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_133.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-61tbxr` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_122.png) Solve an RL circuit to find the current ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/19.png): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_134.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-kzueyy` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_123.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_135.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-xt58sb` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_124.png) Verify with [DSolveValue](https://reference.wolfram.com/language/ref/DSolveValue.html): Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_136.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-z2p4b4` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_125.png) Green's function for an RL circuit: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_137.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-kllajx` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_126.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_138.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-3r61x2` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_127.png) Use the Green's function to solve the RL circuit: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_139.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-10tpdo` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_128.png) Solve a system of ODEs: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_140.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-zm5oii` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_129.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_141.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-5tqdr8` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_130.png) ### Fractional Differential Equations (3) Solve a fractional-order differential equation using Laplace transforms: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_142.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-5yhytr` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_131.png) Solve for the Laplace transform: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_143.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-8g7evs` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_132.png) Find the inverse transform: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_144.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-203z1c` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_133.png) Plot the solution: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_145.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-oepin2` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_134.png) Find the solution directly using [DSolve](https://reference.wolfram.com/language/ref/DSolve.html): Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_146.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-iiovhh` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_135.png) Solve the following fractional integro-differential equation: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_147.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-sboa2x` Direct link to example Solve for the Laplace transform: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_148.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-6hxn7n` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_136.png) Find the inverse transform: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_149.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-ik5ypj` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_137.png) Find the solution directly using [DSolve](https://reference.wolfram.com/language/ref/DSolve.html): Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_150.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-hvse9n` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_138.png) The following equation describes a fractional harmonic oscillator of order 1.9: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_151.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-usm5e4` Direct link to example Solve for the Laplace transform: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_152.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-zsldvm` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_139.png) Find the inverse transform: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_153.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-x8hdfh` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_140.png) Plot the solution: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_154.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-msydxu` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_141.png) Find the solution directly using [DSolve](https://reference.wolfram.com/language/ref/DSolve.html): Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_155.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-ogzgeo` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_142.png) ### Evaluation of Integrals (2) Calculate the following integral: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_156.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-79kl21` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_143.png) Compute the Laplace transform and interchange the order of Laplace transform and integration: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_157.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-t5iy5u` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_144.png) Perform the integration over ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/20.png): Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_158.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-l6recg` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_145.png) Use [InverseLaplaceTransform](https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html) to obtain the original integral: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_159.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-4fo1vb` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_146.png) Verify the result: Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_160.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-feq5fl` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_147.png) Integral involving the Bessel function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_161.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-9dc9dg` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_148.png) Perform a change of variables ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/21.png) and introduce an auxiliary variable ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/22.png): Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_162.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-7gvszw` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_149.png) Apply the Laplace transform and interchange the order of Laplace transform and integration: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_163.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-7aox3f` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_150.png) Perform the integration over ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/23.png): Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_164.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-f0ke5v` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_151.png) Use [InverseLaplaceTransform](https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html) to obtain ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/24.png): Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_165.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-kks51` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_152.png) The original integral equals ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/25.png): Copy to clipboard. In\[6\]:=6 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_166.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-9h7nq4` Direct link to example Out\[6\]=6 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_153.png) Verify the result: Copy to clipboard. In\[7\]:=7 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_167.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-v81car` Direct link to example Out\[7\]=7 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_154.png) ### Other Applications (2) Compute a Laplace transform using a series expansion: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_168.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-wt994` Direct link to example The odd coefficients vanish: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_169.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-699ns9` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_155.png) The transformed series can be summed using [Regularization](https://reference.wolfram.com/language/ref/Regularization.html): Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_170.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-eivhga` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_156.png) Verify the result directly using [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html): Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_171.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-dvmuze` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_157.png) Laplace transform of [Sinc](https://reference.wolfram.com/language/ref/Sinc.html) using series expansions: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_172.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-8z42as` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_173.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-19re7` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_158.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_174.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-fr6xz5` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_159.png) Odd coefficients vanish: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_175.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-cv31y4` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_160.png) Verify the result: Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_176.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-onz02o` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_161.png) ## Properties & Relations (3)Properties of the function, and connections to other functions Use [Asymptotic](https://reference.wolfram.com/language/ref/Asymptotic.html) to compute an asymptotic approximation: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_177.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-d4gzmq` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_162.png) [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html) and [InverseLaplaceTransform](https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html) are mutual inverses: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_178.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-bf3zt1` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_163.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_179.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-4dn9t` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_164.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_180.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-it7bfn` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_165.png) Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_181.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-byd6ra` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_166.png) Use [NIntegrate](https://reference.wolfram.com/language/ref/NIntegrate.html) for numerical approximation: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_182.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-f4nw7q` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_167.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_183.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-cganj7` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_168.png) [NIntegrate](https://reference.wolfram.com/language/ref/NIntegrate.html) computes the transform for numeric values of the Laplace parameter s: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_184.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-g1xcl` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_169.png) ## Possible Issues (1)Common pitfalls and unexpected behavior Simplification can be required to get back the original form: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_185.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-od1` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_170.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_186.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-fxk` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_171.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_187.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-nrq` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_172.png) ## Neat Examples (2)Surprising or curious use cases [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html) done in terms of [MeijerG](https://reference.wolfram.com/language/ref/MeijerG.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_188.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-c92cj` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_173.png) Create a table of basic Laplace transforms: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_189.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-mxc5sq` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_190.png) ✖ `https://wolfram.com/xid/0h2pdw27n2-0odb9` Direct link to example ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_174.png) # See Also [InverseLaplaceTransform](https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html) [BilateralLaplaceTransform](https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.html) [FourierTransform](https://reference.wolfram.com/language/ref/FourierTransform.html) [UnilateralConvolve](https://reference.wolfram.com/language/ref/UnilateralConvolve.html) [ZTransform](https://reference.wolfram.com/language/ref/ZTransform.html) [Integrate](https://reference.wolfram.com/language/ref/Integrate.html) [Piecewise](https://reference.wolfram.com/language/ref/Piecewise.html) [TransferFunctionModel](https://reference.wolfram.com/language/ref/TransferFunctionModel.html) [MellinTransform](https://reference.wolfram.com/language/ref/MellinTransform.html) [Asymptotic](https://reference.wolfram.com/language/ref/Asymptotic.html) [FractionalD](https://reference.wolfram.com/language/ref/FractionalD.html) [CaputoD](https://reference.wolfram.com/language/ref/CaputoD.html) # Tech Notes - [Integral Transforms and Related Operations](https://reference.wolfram.com/language/tutorial/Calculus.html#26017) - [Introduction to Fractional Calculus](https://reference.wolfram.com/language/tutorial/FractionalCalculus.html) # Related Guides - [Fractional Calculus](https://reference.wolfram.com/language/guide/FractionalCalculus.html) - [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.html) - [Generalized Functions](https://reference.wolfram.com/language/guide/GeneralizedFunctions.html) - [Signal Transforms](https://reference.wolfram.com/language/guide/SignalTransforms.html) - [Calculus](https://reference.wolfram.com/language/guide/Calculus.html) - [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.html) - [Signal Processing](https://reference.wolfram.com/language/guide/SignalProcessing.html) - [Summation Transforms](https://reference.wolfram.com/language/guide/SummationTransforms.html) # Related Links - [MathWorld](https://mathworld.wolfram.com/LaplaceTransform.html) # History Introduced in 1999 (4.0) \\| [Updated in 2020 (12.2)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn122) ▪ [2023 (13.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn133) Cite this as: Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023). Copy to clipboard. ✖ `Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023).` ![](https://reference.wolfram.com/language-assets-v2/img/CitationCloseButton.svg) #### Text Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023). Copy to clipboard. ✖ `Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023).` #### CMS Wolfram Language. 1999. "LaplaceTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/LaplaceTransform.html. Copy to clipboard. ✖ `Wolfram Language. 1999. "LaplaceTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/LaplaceTransform.html.` #### APA Wolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LaplaceTransform.html Copy to clipboard. ✖ `Wolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LaplaceTransform.html` #### BibTeX @misc{reference.wolfram\_2025\_laplacetransform, author="Wolfram Research", title="{LaplaceTransform}", year="2023", howpublished="\\url{https://reference.wolfram.com/language/ref/LaplaceTransform.html}", note=\[Accessed: 12-April-2026\]} Copy to clipboard. ✖ `@misc{reference.wolfram_2025_laplacetransform, author="Wolfram Research", title="{LaplaceTransform}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/LaplaceTransform.html}", note=[Accessed: 12-April-2026]}` #### BibLaTeX @online{reference.wolfram\_2025\_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https://reference.wolfram.com/language/ref/LaplaceTransform.html}, note=\[Accessed: 12-April-2026\]} Copy to clipboard. ✖ `@online{reference.wolfram_2025_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https://reference.wolfram.com/language/ref/LaplaceTransform.html}, note=[Accessed: 12-April-2026]}` Give Feedback [Top](https://reference.wolfram.com/language/ref/LaplaceTransform.html#top) [Introduction for Programmers](https://www.wolfram.com/language/fast-introduction-for-programmers/en/) [Introductory Book](https://www.wolfram.com/language/elementary-introduction/) [Wolfram Function Repository](http://resources.wolframcloud.com/FunctionRepository/) \\| [Wolfram Data Repository](https://datarepository.wolframcloud.com/) \\| [Wolfram Data Drop](https://datadrop.wolframcloud.com/) \\| [Wolfram Language Products](https://www.wolfram.com/products/) [Top](https://reference.wolfram.com/language/ref/LaplaceTransform.html#top) - 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Readable Markdown	- [See Also](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [InverseLaplaceTransform](https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html) - [BilateralLaplaceTransform](https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.html) - [FourierTransform](https://reference.wolfram.com/language/ref/FourierTransform.html) - [UnilateralConvolve](https://reference.wolfram.com/language/ref/UnilateralConvolve.html) - [ZTransform](https://reference.wolfram.com/language/ref/ZTransform.html) - [Integrate](https://reference.wolfram.com/language/ref/Integrate.html) - [Piecewise](https://reference.wolfram.com/language/ref/Piecewise.html) - [TransferFunctionModel](https://reference.wolfram.com/language/ref/TransferFunctionModel.html) - [MellinTransform](https://reference.wolfram.com/language/ref/MellinTransform.html) - [Asymptotic](https://reference.wolfram.com/language/ref/Asymptotic.html) - [FractionalD](https://reference.wolfram.com/language/ref/FractionalD.html) - [CaputoD](https://reference.wolfram.com/language/ref/CaputoD.html) - [Related Guides](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [Fractional Calculus](https://reference.wolfram.com/language/guide/FractionalCalculus.html) - [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.html) - [Generalized Functions](https://reference.wolfram.com/language/guide/GeneralizedFunctions.html) - [Signal Transforms](https://reference.wolfram.com/language/guide/SignalTransforms.html) - [Calculus](https://reference.wolfram.com/language/guide/Calculus.html) - [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.html) - [Signal Processing](https://reference.wolfram.com/language/guide/SignalProcessing.html) - [Summation Transforms](https://reference.wolfram.com/language/guide/SummationTransforms.html) - [Tech Notes](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [Integral Transforms and Related Operations](https://reference.wolfram.com/language/tutorial/Calculus.html#26017) - [Introduction to Fractional Calculus](https://reference.wolfram.com/language/tutorial/FractionalCalculus.html) - - [See Also](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [InverseLaplaceTransform](https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html) - [BilateralLaplaceTransform](https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.html) - [FourierTransform](https://reference.wolfram.com/language/ref/FourierTransform.html) - [UnilateralConvolve](https://reference.wolfram.com/language/ref/UnilateralConvolve.html) - [ZTransform](https://reference.wolfram.com/language/ref/ZTransform.html) - [Integrate](https://reference.wolfram.com/language/ref/Integrate.html) - [Piecewise](https://reference.wolfram.com/language/ref/Piecewise.html) - [TransferFunctionModel](https://reference.wolfram.com/language/ref/TransferFunctionModel.html) - [MellinTransform](https://reference.wolfram.com/language/ref/MellinTransform.html) - [Asymptotic](https://reference.wolfram.com/language/ref/Asymptotic.html) - [FractionalD](https://reference.wolfram.com/language/ref/FractionalD.html) - [CaputoD](https://reference.wolfram.com/language/ref/CaputoD.html) - [Related Guides](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [Fractional Calculus](https://reference.wolfram.com/language/guide/FractionalCalculus.html) - [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.html) - [Generalized Functions](https://reference.wolfram.com/language/guide/GeneralizedFunctions.html) - [Signal Transforms](https://reference.wolfram.com/language/guide/SignalTransforms.html) - [Calculus](https://reference.wolfram.com/language/guide/Calculus.html) - [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.html) - [Signal Processing](https://reference.wolfram.com/language/guide/SignalProcessing.html) - [Summation Transforms](https://reference.wolfram.com/language/guide/SummationTransforms.html) - [Tech Notes](https://reference.wolfram.com/language/ref/LaplaceTransform.html) - [Integral Transforms and Related Operations](https://reference.wolfram.com/language/tutorial/Calculus.html#26017) - [Introduction to Fractional Calculus](https://reference.wolfram.com/language/tutorial/FractionalCalculus.html) Examples Basic Examples Scope Basic Uses Elementary Functions Special Functions Formal Properties Numerical Evaluation Fractional Calculus Options Assumptions GenerateConditions Principal Value Working Precision Applications Ordinary Differential Equations Fractional Differential Equations Evaluation of Integrals Other Applications Properties & Relations Possible Issues Neat Examples [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html)\[f\[t\],t,s\] Copy to clipboard. `LaplaceTransform[f[t],t,s]` gives the symbolic Laplace transform of f\[t\] in the variable t as F\[s\] in the variable s. [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html)\[f\[t\],t,![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/1OverscriptBox.png)\] Copy to clipboard. `LaplaceTransform[f[t],t,]` gives the numeric Laplace transform at the numerical value ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/2OverscriptBox.png). [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html)\[f\[t1,…,tn\],{t1,…,tn},{s1,…,sn}\] Copy to clipboard. `LaplaceTransform[f[t1,…,tn],{t1,…,tn},{s1,…,sn}]` gives the multidimensional Laplace transform of f\[t1,…,tn\]. ## Details and Options - Laplace transforms are typically used to transform differential and partial differential equations to algebraic equations, solve and then inverse transform back to a solution. - Laplace transforms are also extensively used in control theory and signal processing as a way to represent and manipulate linear systems in the form of transfer functions and transfer matrices. The Laplace transform and its inverse are then a way to transform between the time domain and frequency domain. - The Laplace transform of a function ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/3.png) is defined to be ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/4.png). - The multidimensional Laplace transform is given by ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/5.png). - The integral is computed using numerical methods if the third argument, s, is given a numerical value. - The asymptotic Laplace transform can be computed using [Asymptotic](https://reference.wolfram.com/language/ref/Asymptotic.html). - The Laplace transform of ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/6.png) exists only for complex values of s in a half-plane ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/7.png). - The lower limit of the integral is effectively taken to be ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/9.png), so that the Laplace transform of the Dirac delta function ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/10.png) is equal to 1. [»](https://reference.wolfram.com/language/ref/LaplaceTransform.html#427361964) - The following options can be given: - \| \| \| \| \| \|---\|---\|---\|---\| \| \| [AccuracyGoal](https://reference.wolfram.com/language/ref/AccuracyGoal.html) \| [Automatic](https://reference.wolfram.com/language/ref/Automatic.html) \| digits of absolute accuracy sought \| \| \| [Assumptions](https://reference.wolfram.com/language/ref/Assumptions.html) \| [\$Assumptions](https://reference.wolfram.com/language/ref/$Assumptions.html) \| assumptions to make about parameters \| \| \| [GenerateConditions](https://reference.wolfram.com/language/ref/GenerateConditions.html) \| [False](https://reference.wolfram.com/language/ref/False.html) \| whether to generate answers that involve conditions on parameters \| \| \| [Method](https://reference.wolfram.com/language/ref/Method.html) \| [Automatic](https://reference.wolfram.com/language/ref/Automatic.html) \| method to use \| \| \| [PerformanceGoal](https://reference.wolfram.com/language/ref/PerformanceGoal.html) \| [\$PerformanceGoal](https://reference.wolfram.com/language/ref/$PerformanceGoal.html) \| aspects of performance to optimize \| \| \| [PrecisionGoal](https://reference.wolfram.com/language/ref/PrecisionGoal.html) \| [Automatic](https://reference.wolfram.com/language/ref/Automatic.html) \| digits of precision sought \| \| \| [PrincipalValue](https://reference.wolfram.com/language/ref/PrincipalValue.html) \| [False](https://reference.wolfram.com/language/ref/False.html) \| whether to find Cauchy principal value \| \| \| [WorkingPrecision](https://reference.wolfram.com/language/ref/WorkingPrecision.html) \| [Automatic](https://reference.wolfram.com/language/ref/Automatic.html) \| the precision used in internal computations \| - Use [GenerateConditions](https://reference.wolfram.com/language/ref/GenerateConditions.html)"ConvergenceRegion" to obtain the region of convergence for the Laplace transform. - In [TraditionalForm](https://reference.wolfram.com/language/ref/TraditionalForm.html), [LaplaceTransform](https://reference.wolfram.com/language/ref/LaplaceTransform.html) is output using ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/11.png). [»](https://reference.wolfram.com/language/ref/LaplaceTransform.html#2079194920) ## Examples open all close all ## Basic Examples (4)Summary of the most common use cases ## Scope (67)Survey of the scope of standard use cases ### Basic Uses (4) ### Elementary Functions (13) ### Special Functions (10) ### Piecewise Functions (9) ### Periodic Functions (5) ### Generalized Functions (5) ### Multivariate Functions (9) Bivariate Laplace transform of a constant: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_83.png) `https://wolfram.com/xid/0h2pdw27n2-l51olx` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_75.png) Exponential function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_84.png) `https://wolfram.com/xid/0h2pdw27n2-iftr0v` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_76.png) Power function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_85.png) `https://wolfram.com/xid/0h2pdw27n2-xn08dr` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_77.png) [BesselJ](https://reference.wolfram.com/language/ref/BesselJ.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_86.png) `https://wolfram.com/xid/0h2pdw27n2-pleo5d` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_78.png) Square root: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_87.png) `https://wolfram.com/xid/0h2pdw27n2-wilqmz` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_79.png) Composition of cosine and square root: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_88.png) `https://wolfram.com/xid/0h2pdw27n2-m19040` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_80.png) Laplace transform of a multivariate power function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_89.png) `https://wolfram.com/xid/0h2pdw27n2-jvc1kh` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_81.png) Cosine: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_90.png) `https://wolfram.com/xid/0h2pdw27n2-w44hxj` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_82.png) Logarithm: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_91.png) `https://wolfram.com/xid/0h2pdw27n2-nse2y4` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_83.png) ### Formal Properties (6) ### Numerical Evaluation (3) ### Fractional Calculus (3) ## Options (4)Common values & functionality for each option ### Assumptions (1) Specify the range for a parameter using [Assumptions](https://reference.wolfram.com/language/ref/Assumptions.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_118.png) `https://wolfram.com/xid/0h2pdw27n2-fkqovc` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_108.png) ### GenerateConditions (1) Use [GenerateConditions](https://reference.wolfram.com/language/ref/GenerateConditions.html)\-\>[True](https://reference.wolfram.com/language/ref/True.html) to get parameter conditions for when a result is valid: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_119.png) `https://wolfram.com/xid/0h2pdw27n2-6pnei` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_109.png) ### Principal Value (1) ### Working Precision (1) ## Applications (12)Sample problems that can be solved with this function ### Ordinary Differential Equations (5) ### Fractional Differential Equations (3) Solve a fractional-order differential equation using Laplace transforms: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_142.png) `https://wolfram.com/xid/0h2pdw27n2-5yhytr` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_131.png) Solve for the Laplace transform: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_143.png) `https://wolfram.com/xid/0h2pdw27n2-8g7evs` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_132.png) Find the inverse transform: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_144.png) `https://wolfram.com/xid/0h2pdw27n2-203z1c` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_133.png) Plot the solution: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_145.png) `https://wolfram.com/xid/0h2pdw27n2-oepin2` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_134.png) Find the solution directly using [DSolve](https://reference.wolfram.com/language/ref/DSolve.html): Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_146.png) `https://wolfram.com/xid/0h2pdw27n2-iiovhh` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_135.png) Solve the following fractional integro-differential equation: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_147.png) `https://wolfram.com/xid/0h2pdw27n2-sboa2x` Direct link to example Solve for the Laplace transform: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_148.png) `https://wolfram.com/xid/0h2pdw27n2-6hxn7n` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_136.png) Find the inverse transform: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_149.png) `https://wolfram.com/xid/0h2pdw27n2-ik5ypj` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_137.png) Find the solution directly using [DSolve](https://reference.wolfram.com/language/ref/DSolve.html): Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_150.png) `https://wolfram.com/xid/0h2pdw27n2-hvse9n` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_138.png) The following equation describes a fractional harmonic oscillator of order 1.9: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_151.png) `https://wolfram.com/xid/0h2pdw27n2-usm5e4` Direct link to example Solve for the Laplace transform: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_152.png) `https://wolfram.com/xid/0h2pdw27n2-zsldvm` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_139.png) Find the inverse transform: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_153.png) `https://wolfram.com/xid/0h2pdw27n2-x8hdfh` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_140.png) Plot the solution: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_154.png) `https://wolfram.com/xid/0h2pdw27n2-msydxu` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_141.png) Find the solution directly using [DSolve](https://reference.wolfram.com/language/ref/DSolve.html): Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/I_155.png) `https://wolfram.com/xid/0h2pdw27n2-ogzgeo` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/LaplaceTransform.en/O_142.png) ### Evaluation of Integrals (2) ### Other Applications (2) ## Properties & Relations (3)Properties of the function, and connections to other functions ## Possible Issues (1)Common pitfalls and unexpected behavior ## Neat Examples (2)Surprising or curious use cases Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023). Copy to clipboard. `Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023).` #### Text Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023). Copy to clipboard. `Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023).` #### CMS Wolfram Language. 1999. "LaplaceTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/LaplaceTransform.html. Copy to clipboard. `Wolfram Language. 1999. "LaplaceTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/LaplaceTransform.html.` #### APA Wolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LaplaceTransform.html Copy to clipboard. `Wolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LaplaceTransform.html` #### BibTeX @misc{reference.wolfram\_2025\_laplacetransform, author="Wolfram Research", title="{LaplaceTransform}", year="2023", howpublished="\\url{https://reference.wolfram.com/language/ref/LaplaceTransform.html}", note=\[Accessed: 12-April-2026\]} Copy to clipboard. `@misc{reference.wolfram_2025_laplacetransform, author="Wolfram Research", title="{LaplaceTransform}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/LaplaceTransform.html}", note=[Accessed: 12-April-2026]}` #### BibLaTeX @online{reference.wolfram\_2025\_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https://reference.wolfram.com/language/ref/LaplaceTransform.html}, note=\[Accessed: 12-April-2026\]} Copy to clipboard. `@online{reference.wolfram_2025_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https://reference.wolfram.com/language/ref/LaplaceTransform.html}, note=[Accessed: 12-April-2026]}`
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