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Calculated Shard: 184 (from laksa142)

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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://mathworld.wolfram.com/LaplaceTransform.html\')), getAhrefsUnparsedNoserviceFromURL(\'https://mathworld.wolfram.com/LaplaceTransform.html\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/mathworld.wolfram.com\/LaplaceTransform.html","crawl_time":1776097884,"first_indexed_time":1562518664,"http_code":200,"src_unparsed":"com,wolfram!mathworld,\/LaplaceTransform.html s443","src_root_hash":"3744487911316863784","history_drop_reason":null,"meta_title":"Laplace Transform -- from Wolfram MathWorld","meta_descriptions":["The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits. The (unilateral) Laplace transform L (not to be confused with the Lie derivative, also commonly denoted L) is defined by L_t[f(t)](s)=int_0^inftyf(t)e^(-st)dt, (1) where f(t) is defined for t>=0..."],"attrs_boilerpipe_text":"Algebra\nApplied Mathematics\nCalculus and Analysis\nDiscrete Mathematics\nFoundations of Mathematics\nGeometry\nHistory and Terminology\nNumber Theory\nProbability and Statistics\nRecreational Mathematics\nTopology\nAlphabetical Index\nNew in MathWorld\nThe Laplace transform is an\nintegral transform\nperhaps second only to the\nFourier transform\nin its utility in solving physical problems. The Laplace transform is particularly\n useful in solving linear\nordinary differential\n equations\nsuch as those arising in the analysis of electronic circuits.\nThe (unilateral) Laplace transform\n(not to be confused with the\nLie\n derivative\n, also commonly denoted\n) is defined by\n(1)\nwhere\nis defined for\n(Abramowitz and Stegun 1972). The unilateral Laplace transform is almost always what\n is meant by \"the\" Laplace transform, although a\nbilateral\n Laplace transform\nis sometimes also defined as\n(2)\n(Oppenheim\net al.\n1997). The unilateral Laplace transform\nis implemented in the\nWolfram\n Language\nas\nLaplaceTransform\n[\nf[t]\n,\nt\n,\ns\n] and the inverse Laplace transform as\nInverseRadonTransform\n.\nThe inverse Laplace transform is known as the\nBromwich integral\n, sometimes known as the Fourier-Mellin integral (see also the related\n Duhamel's convolution principle).\nA table of several important one-sided Laplace transforms is given below.\nconditions\n1\nIn the above table,\nis the zeroth-order\nBessel function\n of the first kind\n,\nis the\ndelta function\n,\n and\nis the\nHeaviside step function\n.\nThe Laplace transform has many important properties. The Laplace transform existence theorem states that, if\nis\npiecewise continuous\non every finite interval in\nsatisfying\n(3)\nfor all\n,\n then\nexists for all\n.\n The Laplace transform is also\nunique\n, in the sense that,\n given two functions\nand\nwith the same transform so that\n(4)\nthen\nLerch's theorem\nguarantees that the integral\n(5)\nvanishes for all\nfor a\nnull function\ndefined by\n(6)\nThe Laplace transform is\nlinear\nsince\n(7)\n(8)\n(9)\nThe Laplace transform of a\nconvolution\nis given by\n(10)\nNow consider\ndifferentiation\n. Let\nbe continuously differentiable\ntimes in\n. If\n, then\n(11)\nThis can be proved by\nintegration by parts\n,\n(12)\n(13)\n(14)\n(15)\nContinuing for higher-order derivatives then gives\n(16)\nThis property can be used to transform differential equations into algebraic equations, a procedure known as the\nHeaviside calculus\n,\n which can then be inverse transformed to obtain the solution. For example, applying\n the Laplace transform to the equation\n(17)\ngives\n(18)\n(19)\nwhich can be rearranged to\n(20)\nIf this equation can be inverse Laplace transformed, then the original differential equation is solved.\nThe Laplace transform satisfied a number of useful properties. Consider\nexponentiation\n. If\nfor\n(i.e.,\nis the Laplace transform of\n), then\nfor\n. This follows from\n(21)\n(22)\n(23)\nThe Laplace transform also has nice properties when applied to\nintegrals\nof functions. If\nis\npiecewise continuous\nand\n, then\n(24)\nSee also\nBilateral Laplace Transform\n,\nBromwich Integral\n,\nFourier-Mellin\n Integral\n,\nFourier Transform\n,\nIntegral\n Transform\n,\nLaplace-Stieltjes Transform\n,\nOperational Mathematics\n,\nUnilateral\n Laplace Transform\nExplore\n this topic in the MathWorld classroom\nExplore with Wolfram|Alpha\nReferences\nAbramowitz, M. and Stegun, I. A. (Eds.). \"Laplace Transforms.\" Ch. 29 in\nHandbook\n of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.\nNew York: Dover, pp. 1019-1030, 1972.\nArfken, G.\nMathematical\n Methods for Physicists, 3rd ed.\nOrlando, FL: Academic Press, pp. 824-863,\n 1985.\nChurchill, R. V.\nOperational\n Mathematics.\nNew York: McGraw-Hill, 1958.\nDoetsch, G.\nIntroduction\n to the Theory and Application of the Laplace Transformation.\nBerlin: Springer-Verlag,\n 1974.\nFranklin, P.\nAn\n Introduction to Fourier Methods and the Laplace Transformation.\nNew York:\n Dover, 1958.\nGraf, U.\nApplied\n Laplace Transforms and\nz\n-Transforms for Scientists and Engineers: A Computational\n Approach using\na\nMathematica Package.\nBasel, Switzerland: Birkhäuser,\n 2004.\nJaeger, J. C. and Newstead, G. H.\nAn\n Introduction to the Laplace Transformation with Engineering Applications.\nLondon: Methuen, 1949.\nHenrici, P.\nApplied\n and Computational Complex Analysis, Vol. 2: Special Functions, Integral Transforms,\n Asymptotics, Continued Fractions.\nNew York: Wiley, pp. 322-350, 1991.\nKrantz,\n S. G. \"The Laplace Transform.\" §15.3 in\nHandbook\n of Complex Variables.\nBoston, MA: Birkhäuser, pp. 212-214, 1999.\nMorse,\n P. M. and Feshbach, H.\nMethods\n of Theoretical Physics, Part I.\nNew York: McGraw-Hill, pp. 467-469,\n 1953.\nOberhettinger, F.\nTables\n of Laplace Transforms.\nNew York: Springer-Verlag, 1973.\nOppenheim,\n A. V.; Willsky, A. S.; and Nawab, S. H.\nSignals\n and Systems, 2nd ed.\nUpper Saddle River, NJ: Prentice-Hall, 1997.\nPrudnikov,\n A. P.; Brychkov, Yu. A.; and Marichev, O. I.\nIntegrals\n and Series, Vol. 4: Direct Laplace Transforms.\nNew York: Gordon and\n Breach, 1992.\nPrudnikov, A. P.; Brychkov, Yu. A.; and Marichev,\n O. I.\nIntegrals\n and Series, Vol. 5: Inverse Laplace Transforms.\nNew York: Gordon and\n Breach, 1992.\nSpiegel, M. R.\nTheory\n and Problems of Laplace Transforms.\nNew York: McGraw-Hill, 1965.\nWeisstein,\n E. W. \"Books about Laplace Transforms.\"\nhttp:\/\/www.ericweisstein.com\/encyclopedias\/books\/LaplaceTransforms.html\n.\nWidder,\n D. V.\nThe\n Laplace Transform.\nPrinceton, NJ: Princeton University Press, 1941.\nZwillinger,\n D. (Ed.).\nCRC\n Standard Mathematical Tables and Formulae.\nBoca Raton, FL: CRC Press, pp. 231\n and 543, 1995.\nReferenced on Wolfram|Alpha\nLaplace Transform\nCite this as:\nWeisstein, Eric W.\n\"Laplace Transform.\"\nFrom\nMathWorld\n--A Wolfram Resource.\nhttps:\/\/mathworld.wolfram.com\/LaplaceTransform.html\nSubject classifications","attrs_markdown":"![](https:\/\/mathworld.wolfram.com\/images\/header\/menu-icon.png) TOPICS\n\n![](https:\/\/mathworld.wolfram.com\/images\/sidebar\/menu-close.png)\n\n[Algebra](https:\/\/mathworld.wolfram.com\/topics\/Algebra.html)   [Applied Mathematics](https:\/\/mathworld.wolfram.com\/topics\/AppliedMathematics.html)   [Calculus and Analysis](https:\/\/mathworld.wolfram.com\/topics\/CalculusandAnalysis.html)   [Discrete Mathematics](https:\/\/mathworld.wolfram.com\/topics\/DiscreteMathematics.html)   [Foundations of Mathematics](https:\/\/mathworld.wolfram.com\/topics\/FoundationsofMathematics.html)   [Geometry](https:\/\/mathworld.wolfram.com\/topics\/Geometry.html)   [History and Terminology](https:\/\/mathworld.wolfram.com\/topics\/HistoryandTerminology.html)   [Number Theory](https:\/\/mathworld.wolfram.com\/topics\/NumberTheory.html)   [Probability and Statistics](https:\/\/mathworld.wolfram.com\/topics\/ProbabilityandStatistics.html)   [Recreational Mathematics](https:\/\/mathworld.wolfram.com\/topics\/RecreationalMathematics.html)   [Topology](https:\/\/mathworld.wolfram.com\/topics\/Topology.html)\n\n[Alphabetical Index](https:\/\/mathworld.wolfram.com\/letters\/)   [New in MathWorld](https:\/\/mathworld.wolfram.com\/whatsnew\/)\n\n- [Calculus and Analysis](https:\/\/mathworld.wolfram.com\/topics\/CalculusandAnalysis.html)\n- [Integral Transforms](https:\/\/mathworld.wolfram.com\/topics\/IntegralTransforms.html)\n- [General Integral Transforms](https:\/\/mathworld.wolfram.com\/topics\/GeneralIntegralTransforms.html)\n\n- [History and Terminology](https:\/\/mathworld.wolfram.com\/topics\/HistoryandTerminology.html)\n- [Wolfram Language Commands](https:\/\/mathworld.wolfram.com\/topics\/WolframLanguageCommands.html)\n# Laplace Transform\n***\n[![DOWNLOAD Mathematica Notebook](https:\/\/mathworld.wolfram.com\/images\/entries\/download-notebook-icon.png)Download Wolfram Notebook](https:\/\/mathworld.wolfram.com\/notebooks\/IntegralTransforms\/LaplaceTransform.nb)\n\nThe Laplace transform is an [integral transform](https:\/\/mathworld.wolfram.com\/IntegralTransform.html) perhaps second only to the [Fourier transform](https:\/\/mathworld.wolfram.com\/FourierTransform.html) in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear [ordinary differential equations](https:\/\/mathworld.wolfram.com\/OrdinaryDifferentialEquation.html) such as those arising in the analysis of electronic circuits.\n\nThe (unilateral) Laplace transform ![L](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline1.svg) (not to be confused with the [Lie derivative](https:\/\/mathworld.wolfram.com\/LieDerivative.html), also commonly denoted ![L](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline2.svg)) is defined by\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[f(t)\\](s)=int\\_0^inftyf(t)e^(-st)dt, ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation1.svg) | (1) |\n\nwhere ![f(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline3.svg) is defined for ![t\\>=0](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline4.svg) (Abramowitz and Stegun 1972). The unilateral Laplace transform is almost always what is meant by \"the\" Laplace transform, although a [bilateral Laplace transform](https:\/\/mathworld.wolfram.com\/BilateralLaplaceTransform.html) is sometimes also defined as\n\n|   |   |\n|---|---|\n| ![ L\\_t^((2))\\[f(t)\\](s)=int\\_(-infty)^inftyf(t)e^(-st)dt ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation2.svg) | (2) |\n\n(Oppenheim *et al.* 1997\\). The unilateral Laplace transform ![L\\_t\\[f(t)\\](s)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline5.svg) is implemented in the [Wolfram Language](http:\/\/www.wolfram.com\/language\/) as [LaplaceTransform](http:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\\[*f\\[t\\]*, *t*, *s*\\] and the inverse Laplace transform as [InverseRadonTransform](http:\/\/reference.wolfram.com\/language\/ref\/InverseRadonTransform.html).\n\nThe inverse Laplace transform is known as the [Bromwich integral](https:\/\/mathworld.wolfram.com\/BromwichIntegral.html), sometimes known as the Fourier-Mellin integral (see also the related Duhamel's convolution principle).\n\nA table of several important one-sided Laplace transforms is given below.\n\n|   |   |   |\n|---|---|---|\n| ![f](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline6.svg) | ![L\\_t\\[f(t)\\](s)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline7.svg) | conditions |\n| 1 | ![1\/s](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline8.svg) |   |\n| ![t](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline9.svg) | ![1\/(s^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline10.svg) |   |\n| ![t^n](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline11.svg) | ![(n!)\/(s^(n+1))](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline12.svg) | ![n in Z\\>=0](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline13.svg) |\n| ![t^a](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline14.svg) | ![(Gamma(a+1))\/(s^(a+1))](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline15.svg) | ![R\\[a\\]\\>-1](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline16.svg) |\n| ![e^(at)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline17.svg) | ![1\/(s-a)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline18.svg) |   |\n| ![cos(omegat)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline19.svg) | ![s\/(s^2+omega^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline20.svg) | ![omega in R](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline21.svg) |\n| ![sin(omegat)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline22.svg) | ![omega\/(s^2+omega^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline23.svg) | ![s\\>\\|I\\[omega\\]\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline24.svg) |\n| ![cosh(omegat)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline25.svg) | ![s\/(s^2-omega^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline26.svg) | ![s\\>\\|R\\[omega\\]\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline27.svg) |\n| ![sinh(omegat)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline28.svg) | ![omega\/(s^2-omega^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline29.svg) | ![s\\>\\|I\\[omega\\]\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline30.svg) |\n| ![e^(at)sin(bt)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline31.svg) | ![b\/((s-a)^2+b^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline32.svg) | ![s\\>a+\\|I\\[b\\]\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline33.svg) |\n| ![e^(at)cos(bt)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline34.svg) | ![(s-a)\/((s-a)^2+b^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline35.svg) | ![b in R](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline36.svg) |\n| ![delta(t-c)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline37.svg) | ![e^(-cs)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline38.svg) |   |\n| ![H\\_c(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline39.svg) | ![{1\/s for c\\<=0; (e^(-cs))\/s for c\\>0](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline40.svg) |   |\n| ![J\\_0(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline41.svg) | ![1\/(sqrt(s^2+1))](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline42.svg) |   |\n| ![J\\_n(at)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline43.svg) | ![((sqrt(s^2+a^2)-s)^n)\/(a^nsqrt(s^2+a^2))](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline44.svg) | ![n in Z\\>=0](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline45.svg) |\n\nIn the above table, ![J\\_0(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline46.svg) is the zeroth-order [Bessel function of the first kind](https:\/\/mathworld.wolfram.com\/BesselFunctionoftheFirstKind.html), ![delta(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline47.svg) is the [delta function](https:\/\/mathworld.wolfram.com\/DeltaFunction.html), and ![H\\_c(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline48.svg) is the [Heaviside step function](https:\/\/mathworld.wolfram.com\/HeavisideStepFunction.html).\n\nThe Laplace transform has many important properties. The Laplace transform existence theorem states that, if ![f(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline49.svg) is [piecewise continuous](https:\/\/mathworld.wolfram.com\/PiecewiseContinuous.html) on every finite interval in ![\\[0,infty)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline50.svg) satisfying\n\n|   |   |\n|---|---|\n| ![ \\|f(t)\\|\\<=Me^(at) ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation3.svg) | (3) |\n\nfor all ![t in \\[0,infty)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline51.svg), then ![L\\_t\\[f(t)\\](s)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline52.svg) exists for all ![s\\>a](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline53.svg). The Laplace transform is also [unique](https:\/\/mathworld.wolfram.com\/Unique.html), in the sense that, given two functions ![F\\_1(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline54.svg) and ![F\\_2(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline55.svg) with the same transform so that\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[F\\_1(t)\\](s)=L\\_t\\[F\\_2(t)\\](s)=f(s), ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation4.svg) | (4) |\n\nthen [Lerch's theorem](https:\/\/mathworld.wolfram.com\/LerchsTheorem.html) guarantees that the integral\n\n|   |   |\n|---|---|\n| ![ int\\_0^aN(t)dt=0 ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation5.svg) | (5) |\n\nvanishes for all ![a\\>0](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline56.svg) for a [null function](https:\/\/mathworld.wolfram.com\/NullFunction.html) defined by\n\n|   |   |\n|---|---|\n| ![ N(t)=F\\_1(t)-F\\_2(t). ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation6.svg) | (6) |\n\nThe Laplace transform is [linear](https:\/\/mathworld.wolfram.com\/LinearOperator.html) since\n\n|   |   |   |   |\n|---|---|---|---|\n| ![L\\_t\\[af(t)+bg(t)\\]](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline57.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline58.svg) | ![int\\_0^infty\\[af(t)+bg(t)\\]e^(-st)dt](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline59.svg) | (7) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline60.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline61.svg) | ![aint\\_0^inftyfe^(-st)dt+bint\\_0^inftyge^(-st)dt](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline62.svg) | (8) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline63.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline64.svg) | ![aL\\_t\\[f(t)\\]+bL\\_t\\[g(t)\\].](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline65.svg) | (9) |\n\nThe Laplace transform of a [convolution](https:\/\/mathworld.wolfram.com\/Convolution.html) is given by\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[f(t)\\*g(t)\\]=L\\_t\\[f(t)\\]L\\_t\\[g(t)\\] L\\_t^(-1)\\[FG\\]=L\\_t^(-1)\\[F\\]\\*L\\_t^(-1)\\[G\\]. ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation7.svg) | (10) |\n\nNow consider [differentiation](https:\/\/mathworld.wolfram.com\/Differentiation.html). Let ![f(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline66.svg) be continuously differentiable ![n-1](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline67.svg) times in ![\\[0,infty)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline68.svg). If ![\\|f(t)\\|\\<=Me^(at)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline69.svg), then\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[f^((n))(t)\\](s)=s^nL\\_t\\[f(t)\\]-s^(n-1)f(0)-s^(n-2)f^'(0)-...-f^((n-1))(0). ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation8.svg) | (11) |\n\nThis can be proved by [integration by parts](https:\/\/mathworld.wolfram.com\/IntegrationbyParts.html),\n\n|   |   |   |   |\n|---|---|---|---|\n| ![L\\_t\\[f^'(t)\\](s)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline70.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline71.svg) | ![lim\\_(a-\\>infty)int\\_0^ae^(-st)f^'(t)dt](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline72.svg) | (12) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline73.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline74.svg) | ![lim\\_(a-\\>infty){\\[e^(-st)f(t)\\]\\_0^a+sint\\_0^ae^(-st)f(t)dt}](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline75.svg) | (13) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline76.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline77.svg) | ![lim\\_(a-\\>infty)\\[e^(-sa)f(a)-f(0)+sint\\_0^ae^(-st)f(t)dt\\]](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline78.svg) | (14) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline79.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline80.svg) | ![sL\\_t\\[f(t)\\]-f(0).](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline81.svg) | (15) |\n\nContinuing for higher-order derivatives then gives\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[f^('')(t)\\](s)=s^2L\\_t\\[f(t)\\](s)-sf(0)-f^'(0). ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation9.svg) | (16) |\n\nThis property can be used to transform differential equations into algebraic equations, a procedure known as the [Heaviside calculus](https:\/\/mathworld.wolfram.com\/HeavisideCalculus.html), which can then be inverse transformed to obtain the solution. For example, applying the Laplace transform to the equation\n\n|   |   |\n|---|---|\n| ![ f^('')(t)+a\\_1f^'(t)+a\\_0f(t)=0 ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation10.svg) | (17) |\n\ngives\n\n|   |   |\n|---|---|\n| ![ {s^2L\\_t\\[f(t)\\](s)-sf(0)-f^'(0)}+a\\_1{sL\\_t\\[f(t)\\](s)-f(0)} +a\\_0L\\_t\\[f(t)\\](s)=0 ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation11.svg) | (18) |\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[f(t)\\](s)(s^2+a\\_1s+a\\_0)-sf(0)-f^'(0)-a\\_1f(0)=0, ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation12.svg) | (19) |\n\nwhich can be rearranged to\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[f(t)\\](s)=(sf(0)+f^'(0)+a\\_1f(0))\/(s^2+a\\_1s+a\\_0). ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation13.svg) | (20) |\n\nIf this equation can be inverse Laplace transformed, then the original differential equation is solved.\n\nThe Laplace transform satisfied a number of useful properties. Consider [exponentiation](https:\/\/mathworld.wolfram.com\/ExponentialFunction.html). If ![L\\_t\\[f(t)\\](s)=F(s)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline82.svg) for ![s\\>alpha](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline83.svg) (i.e., ![F(s)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline84.svg) is the Laplace transform of ![f](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline85.svg)), then ![L\\_t\\[e^(at)f\\](s)=F(s-a)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline86.svg) for ![s\\>a+alpha](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline87.svg). This follows from\n\n|   |   |   |   |\n|---|---|---|---|\n| ![F(s-a)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline88.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline89.svg) | ![int\\_0^inftyfe^(-(s-a)t)dt](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline90.svg) | (21) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline91.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline92.svg) | ![int\\_0^infty\\[f(t)e^(at)\\]e^(-st)dt](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline93.svg) | (22) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline94.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline95.svg) | ![L\\_t\\[e^(at)f(t)\\](s).](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline96.svg) | (23) |\n\nThe Laplace transform also has nice properties when applied to [integrals](https:\/\/mathworld.wolfram.com\/Integral.html) of functions. If ![f(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline97.svg) is [piecewise continuous](https:\/\/mathworld.wolfram.com\/PiecewiseContinuous.html) and ![\\|f(t)\\|\\<=Me^(at)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline98.svg), then\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[int\\_0^tf(t^')dt^'\\]=1\/sL\\_t\\[f(t)\\](s). ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation14.svg) | (24) |\n***\n## See also\n[Bilateral Laplace Transform](https:\/\/mathworld.wolfram.com\/BilateralLaplaceTransform.html), [Bromwich Integral](https:\/\/mathworld.wolfram.com\/BromwichIntegral.html), [Fourier-Mellin Integral](https:\/\/mathworld.wolfram.com\/Fourier-MellinIntegral.html), [Fourier Transform](https:\/\/mathworld.wolfram.com\/FourierTransform.html), [Integral Transform](https:\/\/mathworld.wolfram.com\/IntegralTransform.html), [Laplace-Stieltjes Transform](https:\/\/mathworld.wolfram.com\/Laplace-StieltjesTransform.html), [Operational Mathematics](https:\/\/mathworld.wolfram.com\/OperationalMathematics.html), [Unilateral Laplace Transform](https:\/\/mathworld.wolfram.com\/UnilateralLaplaceTransform.html) [Explore this topic in the MathWorld classroom](https:\/\/mathworld.wolfram.com\/classroom\/LaplaceTransform.html)\n\n## Explore with Wolfram\\|Alpha\n![WolframAlpha](https:\/\/mathworld.wolfram.com\/images\/wolframalpha\/WA-logo.png)\n\nMore things to try:\n\n- [vector field](https:\/\/www.wolframalpha.com\/input\/?i=vector+field)\n- [z-score](https:\/\/www.wolframalpha.com\/input\/?i=z-score)\n- [Laplace transform 1](https:\/\/www.wolframalpha.com\/input\/?i=Laplace+transform+1)\n## References\nAbramowitz, M. and Stegun, I. A. (Eds.). \"Laplace Transforms.\" Ch. 29 in *[Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0486612724\/ref=nosim\/ericstreasuretro)* New York: Dover, pp. 1019-1030, 1972.\n\nArfken, G. *[Mathematical Methods for Physicists, 3rd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0120598760\/ref=nosim\/ericstreasuretro)* Orlando, FL: Academic Press, pp. 824-863, 1985.\n\nChurchill, R. V. *[Operational Mathematics.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0070108706\/ref=nosim\/ericstreasuretro)* New York: McGraw-Hill, 1958.\n\nDoetsch, G. *[Introduction to the Theory and Application of the Laplace Transformation.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0387064079\/ref=nosim\/ericstreasuretro)* Berlin: Springer-Verlag, 1974.\n\nFranklin, P. *[An Introduction to Fourier Methods and the Laplace Transformation.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/B0007HQ89I\/ref=nosim\/ericstreasuretro)* New York: Dover, 1958.\n\nGraf, U.\n\n[Applied Laplace Transforms and *z*\\-Transforms for Scientists and Engineers: A Computational Approach using *a* Mathematica Package.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/3764324279\/ref=nosim\/weisstein-20)\n\nBasel, Switzerland: Birkhäuser, 2004.\n\nJaeger, J. C. and Newstead, G. H. *[An Introduction to the Laplace Transformation with Engineering Applications.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/041612870X\/ref=nosim\/ericstreasuretro)* London: Methuen, 1949.\n\nHenrici, P. *[Applied and Computational Complex Analysis, Vol. 2: Special Functions, Integral Transforms, Asymptotics, Continued Fractions.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/047154289X\/ref=nosim\/ericstreasuretro)* New York: Wiley, pp. 322-350, 1991.\n\nKrantz, S. G. \"The Laplace Transform.\" §15.3 in *[Handbook of Complex Variables.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0817640118\/ref=nosim\/ericstreasuretro)* Boston, MA: Birkhäuser, pp. 212-214, 1999.\n\nMorse, P. M. and Feshbach, H. *[Methods of Theoretical Physics, Part I.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/007043316X\/ref=nosim\/ericstreasuretro)* New York: McGraw-Hill, pp. 467-469, 1953.\n\nOberhettinger, F. *[Tables of Laplace Transforms.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0387063501\/ref=nosim\/ericstreasuretro)* New York: Springer-Verlag, 1973.\n\nOppenheim, A. V.; Willsky, A. S.; and Nawab, S. H. *[Signals and Systems, 2nd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0138147574\/ref=nosim\/ericstreasuretro)* Upper Saddle River, NJ: Prentice-Hall, 1997.\n\nPrudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. *[Integrals and Series, Vol. 4: Direct Laplace Transforms.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/2881248373\/ref=nosim\/ericstreasuretro)* New York: Gordon and Breach, 1992.\n\nPrudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. *[Integrals and Series, Vol. 5: Inverse Laplace Transforms.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/2881248381\/ref=nosim\/ericstreasuretro)* New York: Gordon and Breach, 1992.\n\nSpiegel, M. R. *[Theory and Problems of Laplace Transforms.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/007060231X\/ref=nosim\/ericstreasuretro)* New York: McGraw-Hill, 1965.\n\nWeisstein, E. W. \"Books about Laplace Transforms.\" <http:\/\/www.ericweisstein.com\/encyclopedias\/books\/LaplaceTransforms.html>.\n\nWidder, D. V. *[The Laplace Transform.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0691079927\/ref=nosim\/ericstreasuretro)* Princeton, NJ: Princeton University Press, 1941.\n\nZwillinger, D. (Ed.). *[CRC Standard Mathematical Tables and Formulae.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/1584882913\/ref=nosim\/ericstreasuretro)* Boca Raton, FL: CRC Press, pp. 231 and 543, 1995.\n\n## Referenced on Wolfram\\|Alpha\n[Laplace Transform](https:\/\/www.wolframalpha.com\/input\/?i=laplace+transform \"Laplace Transform\")\n\n## Cite this as:\n[Weisstein, Eric W.](https:\/\/mathworld.wolfram.com\/about\/author.html) \"Laplace Transform.\" From [*MathWorld*](https:\/\/mathworld.wolfram.com\/)\\--A Wolfram Resource. <https:\/\/mathworld.wolfram.com\/LaplaceTransform.html>\n\n## Subject classifications\n- [Calculus and Analysis](https:\/\/mathworld.wolfram.com\/topics\/CalculusandAnalysis.html)\n- [Integral Transforms](https:\/\/mathworld.wolfram.com\/topics\/IntegralTransforms.html)\n- [General Integral Transforms](https:\/\/mathworld.wolfram.com\/topics\/GeneralIntegralTransforms.html)\n\n- [History and Terminology](https:\/\/mathworld.wolfram.com\/topics\/HistoryandTerminology.html)\n- [Wolfram Language Commands](https:\/\/mathworld.wolfram.com\/topics\/WolframLanguageCommands.html)\n\n- [About MathWorld](https:\/\/mathworld.wolfram.com\/about\/)\n- [MathWorld Classroom](https:\/\/mathworld.wolfram.com\/classroom\/)\n- [Contribute](https:\/\/mathworld.wolfram.com\/contact\/)\n- [MathWorld Book](https:\/\/www.amazon.com\/exec\/obidos\/ASIN\/1420072218\/ref=nosim\/weisstein-20)\n- [wolfram.com](https:\/\/www.wolfram.com\/)\n\n- [13,311 Entries](https:\/\/mathworld.wolfram.com\/whatsnew\/)\n- [Last Updated: Wed Mar 25 2026](https:\/\/mathworld.wolfram.com\/whatsnew\/)\n- [©1999–2026 Wolfram Research, Inc.](https:\/\/www.wolfram.com\/)\n- [Terms of Use](https:\/\/www.wolfram.com\/legal\/terms\/mathworld.html)\n\n- [![Wolfram](https:\/\/mathworld.wolfram.com\/images\/footer\/wolfram-logo.png)](https:\/\/www.wolfram.com\/)\n- [wolfram.com](https:\/\/www.wolfram.com\/)\n- [Wolfram for Education](https:\/\/www.wolfram.com\/education\/)\n- Created, developed and nurtured by Eric Weisstein at Wolfram Research\n\n*Created, developed and nurtured by Eric Weisstein at Wolfram Research*\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":"[Algebra](https:\/\/mathworld.wolfram.com\/topics\/Algebra.html)   [Applied Mathematics](https:\/\/mathworld.wolfram.com\/topics\/AppliedMathematics.html)   [Calculus and Analysis](https:\/\/mathworld.wolfram.com\/topics\/CalculusandAnalysis.html)   [Discrete Mathematics](https:\/\/mathworld.wolfram.com\/topics\/DiscreteMathematics.html)   [Foundations of Mathematics](https:\/\/mathworld.wolfram.com\/topics\/FoundationsofMathematics.html)   [Geometry](https:\/\/mathworld.wolfram.com\/topics\/Geometry.html)   [History and Terminology](https:\/\/mathworld.wolfram.com\/topics\/HistoryandTerminology.html)   [Number Theory](https:\/\/mathworld.wolfram.com\/topics\/NumberTheory.html)   [Probability and Statistics](https:\/\/mathworld.wolfram.com\/topics\/ProbabilityandStatistics.html)   [Recreational Mathematics](https:\/\/mathworld.wolfram.com\/topics\/RecreationalMathematics.html)   [Topology](https:\/\/mathworld.wolfram.com\/topics\/Topology.html)\n\n[Alphabetical Index](https:\/\/mathworld.wolfram.com\/letters\/)   [New in MathWorld](https:\/\/mathworld.wolfram.com\/whatsnew\/)\n\n***\nThe Laplace transform is an [integral transform](https:\/\/mathworld.wolfram.com\/IntegralTransform.html) perhaps second only to the [Fourier transform](https:\/\/mathworld.wolfram.com\/FourierTransform.html) in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear [ordinary differential equations](https:\/\/mathworld.wolfram.com\/OrdinaryDifferentialEquation.html) such as those arising in the analysis of electronic circuits.\n\nThe (unilateral) Laplace transform ![L](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline1.svg) (not to be confused with the [Lie derivative](https:\/\/mathworld.wolfram.com\/LieDerivative.html), also commonly denoted ![L](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline2.svg)) is defined by\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[f(t)\\](s)=int\\_0^inftyf(t)e^(-st)dt, ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation1.svg) | (1) |\n\nwhere ![f(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline3.svg) is defined for ![t\\>=0](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline4.svg) (Abramowitz and Stegun 1972). The unilateral Laplace transform is almost always what is meant by \"the\" Laplace transform, although a [bilateral Laplace transform](https:\/\/mathworld.wolfram.com\/BilateralLaplaceTransform.html) is sometimes also defined as\n\n|   |   |\n|---|---|\n| ![ L\\_t^((2))\\[f(t)\\](s)=int\\_(-infty)^inftyf(t)e^(-st)dt ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation2.svg) | (2) |\n\n(Oppenheim *et al.* 1997\\). The unilateral Laplace transform ![L\\_t\\[f(t)\\](s)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline5.svg) is implemented in the [Wolfram Language](http:\/\/www.wolfram.com\/language\/) as [LaplaceTransform](http:\/\/reference.wolfram.com\/language\/ref\/LaplaceTransform.html)\\[*f\\[t\\]*, *t*, *s*\\] and the inverse Laplace transform as [InverseRadonTransform](http:\/\/reference.wolfram.com\/language\/ref\/InverseRadonTransform.html).\n\nThe inverse Laplace transform is known as the [Bromwich integral](https:\/\/mathworld.wolfram.com\/BromwichIntegral.html), sometimes known as the Fourier-Mellin integral (see also the related Duhamel's convolution principle).\n\nA table of several important one-sided Laplace transforms is given below.\n\n|   |   |   |\n|---|---|---|\n| ![f](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline6.svg) | ![L\\_t\\[f(t)\\](s)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline7.svg) | conditions |\n| 1 | ![1\/s](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline8.svg) |   |\n| ![t](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline9.svg) | ![1\/(s^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline10.svg) |   |\n| ![t^n](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline11.svg) | ![(n!)\/(s^(n+1))](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline12.svg) | ![n in Z\\>=0](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline13.svg) |\n| ![t^a](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline14.svg) | ![(Gamma(a+1))\/(s^(a+1))](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline15.svg) | ![R\\[a\\]\\>-1](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline16.svg) |\n| ![e^(at)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline17.svg) | ![1\/(s-a)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline18.svg) |   |\n| ![cos(omegat)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline19.svg) | ![s\/(s^2+omega^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline20.svg) | ![omega in R](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline21.svg) |\n| ![sin(omegat)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline22.svg) | ![omega\/(s^2+omega^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline23.svg) | ![s\\>\\|I\\[omega\\]\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline24.svg) |\n| ![cosh(omegat)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline25.svg) | ![s\/(s^2-omega^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline26.svg) | ![s\\>\\|R\\[omega\\]\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline27.svg) |\n| ![sinh(omegat)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline28.svg) | ![omega\/(s^2-omega^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline29.svg) | ![s\\>\\|I\\[omega\\]\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline30.svg) |\n| ![e^(at)sin(bt)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline31.svg) | ![b\/((s-a)^2+b^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline32.svg) | ![s\\>a+\\|I\\[b\\]\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline33.svg) |\n| ![e^(at)cos(bt)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline34.svg) | ![(s-a)\/((s-a)^2+b^2)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline35.svg) | ![b in R](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline36.svg) |\n| ![delta(t-c)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline37.svg) | ![e^(-cs)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline38.svg) |   |\n| ![H\\_c(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline39.svg) | ![{1\/s for c\\<=0; (e^(-cs))\/s for c\\>0](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline40.svg) |   |\n| ![J\\_0(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline41.svg) | ![1\/(sqrt(s^2+1))](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline42.svg) |   |\n| ![J\\_n(at)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline43.svg) | ![((sqrt(s^2+a^2)-s)^n)\/(a^nsqrt(s^2+a^2))](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline44.svg) | ![n in Z\\>=0](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline45.svg) |\n\nIn the above table, ![J\\_0(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline46.svg) is the zeroth-order [Bessel function of the first kind](https:\/\/mathworld.wolfram.com\/BesselFunctionoftheFirstKind.html), ![delta(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline47.svg) is the [delta function](https:\/\/mathworld.wolfram.com\/DeltaFunction.html), and ![H\\_c(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline48.svg) is the [Heaviside step function](https:\/\/mathworld.wolfram.com\/HeavisideStepFunction.html).\n\nThe Laplace transform has many important properties. The Laplace transform existence theorem states that, if ![f(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline49.svg) is [piecewise continuous](https:\/\/mathworld.wolfram.com\/PiecewiseContinuous.html) on every finite interval in ![\\[0,infty)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline50.svg) satisfying\n\n|   |   |\n|---|---|\n| ![ \\|f(t)\\|\\<=Me^(at) ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation3.svg) | (3) |\n\nfor all ![t in \\[0,infty)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline51.svg), then ![L\\_t\\[f(t)\\](s)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline52.svg) exists for all ![s\\>a](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline53.svg). The Laplace transform is also [unique](https:\/\/mathworld.wolfram.com\/Unique.html), in the sense that, given two functions ![F\\_1(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline54.svg) and ![F\\_2(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline55.svg) with the same transform so that\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[F\\_1(t)\\](s)=L\\_t\\[F\\_2(t)\\](s)=f(s), ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation4.svg) | (4) |\n\nthen [Lerch's theorem](https:\/\/mathworld.wolfram.com\/LerchsTheorem.html) guarantees that the integral\n\n|   |   |\n|---|---|\n| ![ int\\_0^aN(t)dt=0 ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation5.svg) | (5) |\n\nvanishes for all ![a\\>0](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline56.svg) for a [null function](https:\/\/mathworld.wolfram.com\/NullFunction.html) defined by\n\n|   |   |\n|---|---|\n| ![ N(t)=F\\_1(t)-F\\_2(t). ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation6.svg) | (6) |\n\nThe Laplace transform is [linear](https:\/\/mathworld.wolfram.com\/LinearOperator.html) since\n\n|   |   |   |   |\n|---|---|---|---|\n| ![L\\_t\\[af(t)+bg(t)\\]](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline57.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline58.svg) | ![int\\_0^infty\\[af(t)+bg(t)\\]e^(-st)dt](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline59.svg) | (7) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline60.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline61.svg) | ![aint\\_0^inftyfe^(-st)dt+bint\\_0^inftyge^(-st)dt](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline62.svg) | (8) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline63.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline64.svg) | ![aL\\_t\\[f(t)\\]+bL\\_t\\[g(t)\\].](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline65.svg) | (9) |\n\nThe Laplace transform of a [convolution](https:\/\/mathworld.wolfram.com\/Convolution.html) is given by\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[f(t)\\*g(t)\\]=L\\_t\\[f(t)\\]L\\_t\\[g(t)\\] L\\_t^(-1)\\[FG\\]=L\\_t^(-1)\\[F\\]\\*L\\_t^(-1)\\[G\\]. ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation7.svg) | (10) |\n\nNow consider [differentiation](https:\/\/mathworld.wolfram.com\/Differentiation.html). Let ![f(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline66.svg) be continuously differentiable ![n-1](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline67.svg) times in ![\\[0,infty)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline68.svg). If ![\\|f(t)\\|\\<=Me^(at)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline69.svg), then\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[f^((n))(t)\\](s)=s^nL\\_t\\[f(t)\\]-s^(n-1)f(0)-s^(n-2)f^'(0)-...-f^((n-1))(0). ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation8.svg) | (11) |\n\nThis can be proved by [integration by parts](https:\/\/mathworld.wolfram.com\/IntegrationbyParts.html),\n\n|   |   |   |   |\n|---|---|---|---|\n| ![L\\_t\\[f^'(t)\\](s)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline70.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline71.svg) | ![lim\\_(a-\\>infty)int\\_0^ae^(-st)f^'(t)dt](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline72.svg) | (12) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline73.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline74.svg) | ![lim\\_(a-\\>infty){\\[e^(-st)f(t)\\]\\_0^a+sint\\_0^ae^(-st)f(t)dt}](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline75.svg) | (13) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline76.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline77.svg) | ![lim\\_(a-\\>infty)\\[e^(-sa)f(a)-f(0)+sint\\_0^ae^(-st)f(t)dt\\]](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline78.svg) | (14) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline79.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline80.svg) | ![sL\\_t\\[f(t)\\]-f(0).](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline81.svg) | (15) |\n\nContinuing for higher-order derivatives then gives\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[f^('')(t)\\](s)=s^2L\\_t\\[f(t)\\](s)-sf(0)-f^'(0). ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation9.svg) | (16) |\n\nThis property can be used to transform differential equations into algebraic equations, a procedure known as the [Heaviside calculus](https:\/\/mathworld.wolfram.com\/HeavisideCalculus.html), which can then be inverse transformed to obtain the solution. For example, applying the Laplace transform to the equation\n\n|   |   |\n|---|---|\n| ![ f^('')(t)+a\\_1f^'(t)+a\\_0f(t)=0 ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation10.svg) | (17) |\n\ngives\n\n|   |   |\n|---|---|\n| ![ {s^2L\\_t\\[f(t)\\](s)-sf(0)-f^'(0)}+a\\_1{sL\\_t\\[f(t)\\](s)-f(0)} +a\\_0L\\_t\\[f(t)\\](s)=0 ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation11.svg) | (18) |\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[f(t)\\](s)(s^2+a\\_1s+a\\_0)-sf(0)-f^'(0)-a\\_1f(0)=0, ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation12.svg) | (19) |\n\nwhich can be rearranged to\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[f(t)\\](s)=(sf(0)+f^'(0)+a\\_1f(0))\/(s^2+a\\_1s+a\\_0). ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation13.svg) | (20) |\n\nIf this equation can be inverse Laplace transformed, then the original differential equation is solved.\n\nThe Laplace transform satisfied a number of useful properties. Consider [exponentiation](https:\/\/mathworld.wolfram.com\/ExponentialFunction.html). If ![L\\_t\\[f(t)\\](s)=F(s)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline82.svg) for ![s\\>alpha](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline83.svg) (i.e., ![F(s)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline84.svg) is the Laplace transform of ![f](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline85.svg)), then ![L\\_t\\[e^(at)f\\](s)=F(s-a)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline86.svg) for ![s\\>a+alpha](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline87.svg). This follows from\n\n|   |   |   |   |\n|---|---|---|---|\n| ![F(s-a)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline88.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline89.svg) | ![int\\_0^inftyfe^(-(s-a)t)dt](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline90.svg) | (21) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline91.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline92.svg) | ![int\\_0^infty\\[f(t)e^(at)\\]e^(-st)dt](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline93.svg) | (22) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline94.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline95.svg) | ![L\\_t\\[e^(at)f(t)\\](s).](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline96.svg) | (23) |\n\nThe Laplace transform also has nice properties when applied to [integrals](https:\/\/mathworld.wolfram.com\/Integral.html) of functions. If ![f(t)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline97.svg) is [piecewise continuous](https:\/\/mathworld.wolfram.com\/PiecewiseContinuous.html) and ![\\|f(t)\\|\\<=Me^(at)](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/Inline98.svg), then\n\n|   |   |\n|---|---|\n| ![ L\\_t\\[int\\_0^tf(t^')dt^'\\]=1\/sL\\_t\\[f(t)\\](s). ](https:\/\/mathworld.wolfram.com\/images\/equations\/LaplaceTransform\/NumberedEquation14.svg) | (24) |\n***\n## See also\n[Bilateral Laplace Transform](https:\/\/mathworld.wolfram.com\/BilateralLaplaceTransform.html), [Bromwich Integral](https:\/\/mathworld.wolfram.com\/BromwichIntegral.html), [Fourier-Mellin Integral](https:\/\/mathworld.wolfram.com\/Fourier-MellinIntegral.html), [Fourier Transform](https:\/\/mathworld.wolfram.com\/FourierTransform.html), [Integral Transform](https:\/\/mathworld.wolfram.com\/IntegralTransform.html), [Laplace-Stieltjes Transform](https:\/\/mathworld.wolfram.com\/Laplace-StieltjesTransform.html), [Operational Mathematics](https:\/\/mathworld.wolfram.com\/OperationalMathematics.html), [Unilateral Laplace Transform](https:\/\/mathworld.wolfram.com\/UnilateralLaplaceTransform.html) [Explore this topic in the MathWorld classroom](https:\/\/mathworld.wolfram.com\/classroom\/LaplaceTransform.html)\n\n## Explore with Wolfram\\|Alpha\n## References\nAbramowitz, M. and Stegun, I. A. (Eds.). \"Laplace Transforms.\" Ch. 29 in *[Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0486612724\/ref=nosim\/ericstreasuretro)* New York: Dover, pp. 1019-1030, 1972.Arfken, G. *[Mathematical Methods for Physicists, 3rd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0120598760\/ref=nosim\/ericstreasuretro)* Orlando, FL: Academic Press, pp. 824-863, 1985.Churchill, R. V. *[Operational Mathematics.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0070108706\/ref=nosim\/ericstreasuretro)* New York: McGraw-Hill, 1958.Doetsch, G. *[Introduction to the Theory and Application of the Laplace Transformation.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0387064079\/ref=nosim\/ericstreasuretro)* Berlin: Springer-Verlag, 1974.Franklin, P. *[An Introduction to Fourier Methods and the Laplace Transformation.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/B0007HQ89I\/ref=nosim\/ericstreasuretro)* New York: Dover, 1958.Graf, U. [Applied Laplace Transforms and *z*\\-Transforms for Scientists and Engineers: A Computational Approach using *a* Mathematica Package.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/3764324279\/ref=nosim\/weisstein-20) Basel, Switzerland: Birkhäuser, 2004.Jaeger, J. C. and Newstead, G. H. *[An Introduction to the Laplace Transformation with Engineering Applications.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/041612870X\/ref=nosim\/ericstreasuretro)* London: Methuen, 1949.Henrici, P. *[Applied and Computational Complex Analysis, Vol. 2: Special Functions, Integral Transforms, Asymptotics, Continued Fractions.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/047154289X\/ref=nosim\/ericstreasuretro)* New York: Wiley, pp. 322-350, 1991.Krantz, S. G. \"The Laplace Transform.\" §15.3 in *[Handbook of Complex Variables.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0817640118\/ref=nosim\/ericstreasuretro)* Boston, MA: Birkhäuser, pp. 212-214, 1999.Morse, P. M. and Feshbach, H. *[Methods of Theoretical Physics, Part I.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/007043316X\/ref=nosim\/ericstreasuretro)* New York: McGraw-Hill, pp. 467-469, 1953.Oberhettinger, F. *[Tables of Laplace Transforms.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0387063501\/ref=nosim\/ericstreasuretro)* New York: Springer-Verlag, 1973.Oppenheim, A. V.; Willsky, A. S.; and Nawab, S. H. *[Signals and Systems, 2nd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0138147574\/ref=nosim\/ericstreasuretro)* Upper Saddle River, NJ: Prentice-Hall, 1997.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. *[Integrals and Series, Vol. 4: Direct Laplace Transforms.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/2881248373\/ref=nosim\/ericstreasuretro)* New York: Gordon and Breach, 1992.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. *[Integrals and Series, Vol. 5: Inverse Laplace Transforms.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/2881248381\/ref=nosim\/ericstreasuretro)* New York: Gordon and Breach, 1992.Spiegel, M. R. *[Theory and Problems of Laplace Transforms.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/007060231X\/ref=nosim\/ericstreasuretro)* New York: McGraw-Hill, 1965.Weisstein, E. W. \"Books about Laplace Transforms.\" <http:\/\/www.ericweisstein.com\/encyclopedias\/books\/LaplaceTransforms.html>.Widder, D. V. *[The Laplace Transform.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0691079927\/ref=nosim\/ericstreasuretro)* Princeton, NJ: Princeton University Press, 1941.Zwillinger, D. (Ed.). *[CRC Standard Mathematical Tables and Formulae.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/1584882913\/ref=nosim\/ericstreasuretro)* Boca Raton, FL: CRC Press, pp. 231 and 543, 1995.\n\n## Referenced on Wolfram\\|Alpha\n[Laplace Transform](https:\/\/www.wolframalpha.com\/input\/?i=laplace+transform \"Laplace Transform\")\n\n## Cite this as:\n[Weisstein, Eric W.](https:\/\/mathworld.wolfram.com\/about\/author.html) \"Laplace Transform.\" From [*MathWorld*](https:\/\/mathworld.wolfram.com\/)\\--A Wolfram Resource. <https:\/\/mathworld.wolfram.com\/LaplaceTransform.html>\n\n## Subject classifications","meta_canonical":null}

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Meta Title	Laplace Transform -- from Wolfram MathWorld
Meta Description	The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits. The (unilateral) Laplace transform L (not to be confused with the Lie derivative, also commonly denoted L) is defined by L_t[f(t)](s)=int_0^inftyf(t)e^(-st)dt, (1) where f(t) is defined for t>=0...
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Boilerpipe Text	Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index New in MathWorld The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits. The (unilateral) Laplace transform (not to be confused with the Lie derivative , also commonly denoted ) is defined by (1) where is defined for (Abramowitz and Stegun 1972). The unilateral Laplace transform is almost always what is meant by "the" Laplace transform, although a bilateral Laplace transform is sometimes also defined as (2) (Oppenheim et al. 1997). The unilateral Laplace transform is implemented in the Wolfram Language as LaplaceTransform [ f[t] , t , s ] and the inverse Laplace transform as InverseRadonTransform . The inverse Laplace transform is known as the Bromwich integral , sometimes known as the Fourier-Mellin integral (see also the related Duhamel's convolution principle). A table of several important one-sided Laplace transforms is given below. conditions 1 In the above table, is the zeroth-order Bessel function of the first kind , is the delta function , and is the Heaviside step function . The Laplace transform has many important properties. The Laplace transform existence theorem states that, if is piecewise continuous on every finite interval in satisfying (3) for all , then exists for all . The Laplace transform is also unique , in the sense that, given two functions and with the same transform so that (4) then Lerch's theorem guarantees that the integral (5) vanishes for all for a null function defined by (6) The Laplace transform is linear since (7) (8) (9) The Laplace transform of a convolution is given by (10) Now consider differentiation . Let be continuously differentiable times in . If , then (11) This can be proved by integration by parts , (12) (13) (14) (15) Continuing for higher-order derivatives then gives (16) This property can be used to transform differential equations into algebraic equations, a procedure known as the Heaviside calculus , which can then be inverse transformed to obtain the solution. For example, applying the Laplace transform to the equation (17) gives (18) (19) which can be rearranged to (20) If this equation can be inverse Laplace transformed, then the original differential equation is solved. The Laplace transform satisfied a number of useful properties. Consider exponentiation . If for (i.e., is the Laplace transform of ), then for . This follows from (21) (22) (23) The Laplace transform also has nice properties when applied to integrals of functions. If is piecewise continuous and , then (24) See also Bilateral Laplace Transform , Bromwich Integral , Fourier-Mellin Integral , Fourier Transform , Integral Transform , Laplace-Stieltjes Transform , Operational Mathematics , Unilateral Laplace Transform Explore this topic in the MathWorld classroom Explore with Wolfram\|Alpha References Abramowitz, M. and Stegun, I. A. (Eds.). "Laplace Transforms." Ch. 29 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1019-1030, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 824-863, 1985. Churchill, R. V. Operational Mathematics. New York: McGraw-Hill, 1958. Doetsch, G. Introduction to the Theory and Application of the Laplace Transformation. Berlin: Springer-Verlag, 1974. Franklin, P. An Introduction to Fourier Methods and the Laplace Transformation. New York: Dover, 1958. Graf, U. Applied Laplace Transforms and z -Transforms for Scientists and Engineers: A Computational Approach using a Mathematica Package. Basel, Switzerland: Birkhäuser, 2004. Jaeger, J. C. and Newstead, G. H. An Introduction to the Laplace Transformation with Engineering Applications. London: Methuen, 1949. Henrici, P. Applied and Computational Complex Analysis, Vol. 2: Special Functions, Integral Transforms, Asymptotics, Continued Fractions. New York: Wiley, pp. 322-350, 1991. Krantz, S. G. "The Laplace Transform." §15.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 212-214, 1999. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 467-469, 1953. Oberhettinger, F. Tables of Laplace Transforms. New York: Springer-Verlag, 1973. Oppenheim, A. V.; Willsky, A. S.; and Nawab, S. H. Signals and Systems, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1997. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 4: Direct Laplace Transforms. New York: Gordon and Breach, 1992. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 5: Inverse Laplace Transforms. New York: Gordon and Breach, 1992. Spiegel, M. R. Theory and Problems of Laplace Transforms. New York: McGraw-Hill, 1965. Weisstein, E. W. "Books about Laplace Transforms." http://www.ericweisstein.com/encyclopedias/books/LaplaceTransforms.html . Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University Press, 1941. Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 231 and 543, 1995. Referenced on Wolfram\|Alpha Laplace Transform Cite this as: Weisstein, Eric W. "Laplace Transform." From MathWorld --A Wolfram Resource. https://mathworld.wolfram.com/LaplaceTransform.html Subject classifications
Markdown	![](https://mathworld.wolfram.com/images/header/menu-icon.png) TOPICS ![](https://mathworld.wolfram.com/images/sidebar/menu-close.png) [Algebra](https://mathworld.wolfram.com/topics/Algebra.html) [Applied Mathematics](https://mathworld.wolfram.com/topics/AppliedMathematics.html) [Calculus and Analysis](https://mathworld.wolfram.com/topics/CalculusandAnalysis.html) [Discrete Mathematics](https://mathworld.wolfram.com/topics/DiscreteMathematics.html) [Foundations of Mathematics](https://mathworld.wolfram.com/topics/FoundationsofMathematics.html) [Geometry](https://mathworld.wolfram.com/topics/Geometry.html) [History and Terminology](https://mathworld.wolfram.com/topics/HistoryandTerminology.html) [Number Theory](https://mathworld.wolfram.com/topics/NumberTheory.html) [Probability and Statistics](https://mathworld.wolfram.com/topics/ProbabilityandStatistics.html) [Recreational Mathematics](https://mathworld.wolfram.com/topics/RecreationalMathematics.html) [Topology](https://mathworld.wolfram.com/topics/Topology.html) [Alphabetical Index](https://mathworld.wolfram.com/letters/) [New in MathWorld](https://mathworld.wolfram.com/whatsnew/) - [Calculus and Analysis](https://mathworld.wolfram.com/topics/CalculusandAnalysis.html) - [Integral Transforms](https://mathworld.wolfram.com/topics/IntegralTransforms.html) - [General Integral Transforms](https://mathworld.wolfram.com/topics/GeneralIntegralTransforms.html) - [History and Terminology](https://mathworld.wolfram.com/topics/HistoryandTerminology.html) - [Wolfram Language Commands](https://mathworld.wolfram.com/topics/WolframLanguageCommands.html) # Laplace Transform *** [![DOWNLOAD Mathematica Notebook](https://mathworld.wolfram.com/images/entries/download-notebook-icon.png)Download Wolfram Notebook](https://mathworld.wolfram.com/notebooks/IntegralTransforms/LaplaceTransform.nb) The Laplace transform is an [integral transform](https://mathworld.wolfram.com/IntegralTransform.html) perhaps second only to the [Fourier transform](https://mathworld.wolfram.com/FourierTransform.html) in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear [ordinary differential equations](https://mathworld.wolfram.com/OrdinaryDifferentialEquation.html) such as those arising in the analysis of electronic circuits. The (unilateral) Laplace transform ![L](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline1.svg) (not to be confused with the [Lie derivative](https://mathworld.wolfram.com/LieDerivative.html), also commonly denoted ![L](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline2.svg)) is defined by \| \| \| \|---\|---\| \| ![ L\_t\[f(t)\](s)=int\_0^inftyf(t)e^(-st)dt, ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation1.svg) \| (1) \| where ![f(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline3.svg) is defined for ![t\>=0](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline4.svg) (Abramowitz and Stegun 1972). The unilateral Laplace transform is almost always what is meant by "the" Laplace transform, although a [bilateral Laplace transform](https://mathworld.wolfram.com/BilateralLaplaceTransform.html) is sometimes also defined as \| \| \| \|---\|---\| \| ![ L\_t^((2))\[f(t)\](s)=int\_(-infty)^inftyf(t)e^(-st)dt ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation2.svg) \| (2) \| (Oppenheim et al. 1997\). The unilateral Laplace transform ![L\_t\[f(t)\](s)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline5.svg) is implemented in the [Wolfram Language](http://www.wolfram.com/language/) as [LaplaceTransform](http://reference.wolfram.com/language/ref/LaplaceTransform.html)\[f\[t\], t, s\] and the inverse Laplace transform as [InverseRadonTransform](http://reference.wolfram.com/language/ref/InverseRadonTransform.html). The inverse Laplace transform is known as the [Bromwich integral](https://mathworld.wolfram.com/BromwichIntegral.html), sometimes known as the Fourier-Mellin integral (see also the related Duhamel's convolution principle). A table of several important one-sided Laplace transforms is given below. \| \| \| \| \|---\|---\|---\| \| ![f](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline6.svg) \| ![L\_t\[f(t)\](s)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline7.svg) \| conditions \| \| 1 \| ![1/s](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline8.svg) \| \| \| ![t](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline9.svg) \| ![1/(s^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline10.svg) \| \| \| ![t^n](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline11.svg) \| ![(n!)/(s^(n+1))](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline12.svg) \| ![n in Z\>=0](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline13.svg) \| \| ![t^a](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline14.svg) \| ![(Gamma(a+1))/(s^(a+1))](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline15.svg) \| ![R\[a\]\>-1](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline16.svg) \| \| ![e^(at)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline17.svg) \| ![1/(s-a)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline18.svg) \| \| \| ![cos(omegat)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline19.svg) \| ![s/(s^2+omega^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline20.svg) \| ![omega in R](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline21.svg) \| \| ![sin(omegat)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline22.svg) \| ![omega/(s^2+omega^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline23.svg) \| ![s\>\\|I\[omega\]\\|](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline24.svg) \| \| ![cosh(omegat)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline25.svg) \| ![s/(s^2-omega^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline26.svg) \| ![s\>\\|R\[omega\]\\|](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline27.svg) \| \| ![sinh(omegat)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline28.svg) \| ![omega/(s^2-omega^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline29.svg) \| ![s\>\\|I\[omega\]\\|](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline30.svg) \| \| ![e^(at)sin(bt)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline31.svg) \| ![b/((s-a)^2+b^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline32.svg) \| ![s\>a+\\|I\[b\]\\|](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline33.svg) \| \| ![e^(at)cos(bt)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline34.svg) \| ![(s-a)/((s-a)^2+b^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline35.svg) \| ![b in R](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline36.svg) \| \| ![delta(t-c)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline37.svg) \| ![e^(-cs)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline38.svg) \| \| \| ![H\_c(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline39.svg) \| ![{1/s for c\<=0; (e^(-cs))/s for c\>0](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline40.svg) \| \| \| ![J\_0(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline41.svg) \| ![1/(sqrt(s^2+1))](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline42.svg) \| \| \| ![J\_n(at)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline43.svg) \| ![((sqrt(s^2+a^2)-s)^n)/(a^nsqrt(s^2+a^2))](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline44.svg) \| ![n in Z\>=0](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline45.svg) \| In the above table, ![J\_0(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline46.svg) is the zeroth-order [Bessel function of the first kind](https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html), ![delta(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline47.svg) is the [delta function](https://mathworld.wolfram.com/DeltaFunction.html), and ![H\_c(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline48.svg) is the [Heaviside step function](https://mathworld.wolfram.com/HeavisideStepFunction.html). The Laplace transform has many important properties. The Laplace transform existence theorem states that, if ![f(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline49.svg) is [piecewise continuous](https://mathworld.wolfram.com/PiecewiseContinuous.html) on every finite interval in ![\[0,infty)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline50.svg) satisfying \| \| \| \|---\|---\| \| ![ \\|f(t)\\|\<=Me^(at) ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation3.svg) \| (3) \| for all ![t in \[0,infty)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline51.svg), then ![L\_t\[f(t)\](s)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline52.svg) exists for all ![s\>a](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline53.svg). The Laplace transform is also [unique](https://mathworld.wolfram.com/Unique.html), in the sense that, given two functions ![F\_1(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline54.svg) and ![F\_2(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline55.svg) with the same transform so that \| \| \| \|---\|---\| \| ![ L\_t\[F\_1(t)\](s)=L\_t\[F\_2(t)\](s)=f(s), ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation4.svg) \| (4) \| then [Lerch's theorem](https://mathworld.wolfram.com/LerchsTheorem.html) guarantees that the integral \| \| \| \|---\|---\| \| ![ int\_0^aN(t)dt=0 ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation5.svg) \| (5) \| vanishes for all ![a\>0](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline56.svg) for a [null function](https://mathworld.wolfram.com/NullFunction.html) defined by \| \| \| \|---\|---\| \| ![ N(t)=F\_1(t)-F\_2(t). ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation6.svg) \| (6) \| The Laplace transform is [linear](https://mathworld.wolfram.com/LinearOperator.html) since \| \| \| \| \| \|---\|---\|---\|---\| \| ![L\_t\[af(t)+bg(t)\]](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline57.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline58.svg) \| ![int\_0^infty\[af(t)+bg(t)\]e^(-st)dt](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline59.svg) \| (7) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline60.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline61.svg) \| ![aint\_0^inftyfe^(-st)dt+bint\_0^inftyge^(-st)dt](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline62.svg) \| (8) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline63.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline64.svg) \| ![aL\_t\[f(t)\]+bL\_t\[g(t)\].](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline65.svg) \| (9) \| The Laplace transform of a [convolution](https://mathworld.wolfram.com/Convolution.html) is given by \| \| \| \|---\|---\| \| ![ L\_t\[f(t)\g(t)\]=L\_t\[f(t)\]L\_t\[g(t)\] L\_t^(-1)\[FG\]=L\_t^(-1)\[F\]\L\_t^(-1)\[G\]. ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation7.svg) \| (10) \| Now consider [differentiation](https://mathworld.wolfram.com/Differentiation.html). Let ![f(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline66.svg) be continuously differentiable ![n-1](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline67.svg) times in ![\[0,infty)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline68.svg). If ![\\|f(t)\\|\<=Me^(at)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline69.svg), then \| \| \| \|---\|---\| \| ![ L\_t\[f^((n))(t)\](s)=s^nL\_t\[f(t)\]-s^(n-1)f(0)-s^(n-2)f^'(0)-...-f^((n-1))(0). ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation8.svg) \| (11) \| This can be proved by [integration by parts](https://mathworld.wolfram.com/IntegrationbyParts.html), \| \| \| \| \| \|---\|---\|---\|---\| \| ![L\_t\[f^'(t)\](s)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline70.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline71.svg) \| ![lim\_(a-\>infty)int\_0^ae^(-st)f^'(t)dt](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline72.svg) \| (12) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline73.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline74.svg) \| ![lim\_(a-\>infty){\[e^(-st)f(t)\]\_0^a+sint\_0^ae^(-st)f(t)dt}](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline75.svg) \| (13) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline76.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline77.svg) \| ![lim\_(a-\>infty)\[e^(-sa)f(a)-f(0)+sint\_0^ae^(-st)f(t)dt\]](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline78.svg) \| (14) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline79.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline80.svg) \| ![sL\_t\[f(t)\]-f(0).](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline81.svg) \| (15) \| Continuing for higher-order derivatives then gives \| \| \| \|---\|---\| \| ![ L\_t\[f^('')(t)\](s)=s^2L\_t\[f(t)\](s)-sf(0)-f^'(0). ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation9.svg) \| (16) \| This property can be used to transform differential equations into algebraic equations, a procedure known as the [Heaviside calculus](https://mathworld.wolfram.com/HeavisideCalculus.html), which can then be inverse transformed to obtain the solution. For example, applying the Laplace transform to the equation \| \| \| \|---\|---\| \| ![ f^('')(t)+a\_1f^'(t)+a\_0f(t)=0 ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation10.svg) \| (17) \| gives \| \| \| \|---\|---\| \| ![ {s^2L\_t\[f(t)\](s)-sf(0)-f^'(0)}+a\_1{sL\_t\[f(t)\](s)-f(0)} +a\_0L\_t\[f(t)\](s)=0 ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation11.svg) \| (18) \| \| \| \| \|---\|---\| \| ![ L\_t\[f(t)\](s)(s^2+a\_1s+a\_0)-sf(0)-f^'(0)-a\_1f(0)=0, ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation12.svg) \| (19) \| which can be rearranged to \| \| \| \|---\|---\| \| ![ L\_t\[f(t)\](s)=(sf(0)+f^'(0)+a\_1f(0))/(s^2+a\_1s+a\_0). ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation13.svg) \| (20) \| If this equation can be inverse Laplace transformed, then the original differential equation is solved. The Laplace transform satisfied a number of useful properties. Consider [exponentiation](https://mathworld.wolfram.com/ExponentialFunction.html). If ![L\_t\[f(t)\](s)=F(s)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline82.svg) for ![s\>alpha](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline83.svg) (i.e., ![F(s)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline84.svg) is the Laplace transform of ![f](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline85.svg)), then ![L\_t\[e^(at)f\](s)=F(s-a)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline86.svg) for ![s\>a+alpha](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline87.svg). This follows from \| \| \| \| \| \|---\|---\|---\|---\| \| ![F(s-a)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline88.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline89.svg) \| ![int\_0^inftyfe^(-(s-a)t)dt](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline90.svg) \| (21) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline91.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline92.svg) \| ![int\_0^infty\[f(t)e^(at)\]e^(-st)dt](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline93.svg) \| (22) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline94.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline95.svg) \| ![L\_t\[e^(at)f(t)\](s).](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline96.svg) \| (23) \| The Laplace transform also has nice properties when applied to [integrals](https://mathworld.wolfram.com/Integral.html) of functions. If ![f(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline97.svg) is [piecewise continuous](https://mathworld.wolfram.com/PiecewiseContinuous.html) and ![\\|f(t)\\|\<=Me^(at)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline98.svg), then \| \| \| \|---\|---\| \| ![ L\_t\[int\_0^tf(t^')dt^'\]=1/sL\_t\[f(t)\](s). ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation14.svg) \| (24) \| *** ## See also [Bilateral Laplace Transform](https://mathworld.wolfram.com/BilateralLaplaceTransform.html), [Bromwich Integral](https://mathworld.wolfram.com/BromwichIntegral.html), [Fourier-Mellin Integral](https://mathworld.wolfram.com/Fourier-MellinIntegral.html), [Fourier Transform](https://mathworld.wolfram.com/FourierTransform.html), [Integral Transform](https://mathworld.wolfram.com/IntegralTransform.html), [Laplace-Stieltjes Transform](https://mathworld.wolfram.com/Laplace-StieltjesTransform.html), [Operational Mathematics](https://mathworld.wolfram.com/OperationalMathematics.html), [Unilateral Laplace Transform](https://mathworld.wolfram.com/UnilateralLaplaceTransform.html) [Explore this topic in the MathWorld classroom](https://mathworld.wolfram.com/classroom/LaplaceTransform.html) ## Explore with Wolfram\\|Alpha ![WolframAlpha](https://mathworld.wolfram.com/images/wolframalpha/WA-logo.png) More things to try: - [vector field](https://www.wolframalpha.com/input/?i=vector+field) - [z-score](https://www.wolframalpha.com/input/?i=z-score) - [Laplace transform 1](https://www.wolframalpha.com/input/?i=Laplace+transform+1) ## References Abramowitz, M. and Stegun, I. A. (Eds.). "Laplace Transforms." Ch. 29 in [Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.](http://www.amazon.com/exec/obidos/ASIN/0486612724/ref=nosim/ericstreasuretro) New York: Dover, pp. 1019-1030, 1972. Arfken, G. [Mathematical Methods for Physicists, 3rd ed.](http://www.amazon.com/exec/obidos/ASIN/0120598760/ref=nosim/ericstreasuretro) Orlando, FL: Academic Press, pp. 824-863, 1985. Churchill, R. V. [Operational Mathematics.](http://www.amazon.com/exec/obidos/ASIN/0070108706/ref=nosim/ericstreasuretro) New York: McGraw-Hill, 1958. Doetsch, G. [Introduction to the Theory and Application of the Laplace Transformation.](http://www.amazon.com/exec/obidos/ASIN/0387064079/ref=nosim/ericstreasuretro) Berlin: Springer-Verlag, 1974. Franklin, P. [An Introduction to Fourier Methods and the Laplace Transformation.](http://www.amazon.com/exec/obidos/ASIN/B0007HQ89I/ref=nosim/ericstreasuretro) New York: Dover, 1958. Graf, U. [Applied Laplace Transforms and z\-Transforms for Scientists and Engineers: A Computational Approach using a Mathematica Package.](http://www.amazon.com/exec/obidos/ASIN/3764324279/ref=nosim/weisstein-20) Basel, Switzerland: Birkhäuser, 2004. Jaeger, J. C. and Newstead, G. H. [An Introduction to the Laplace Transformation with Engineering Applications.](http://www.amazon.com/exec/obidos/ASIN/041612870X/ref=nosim/ericstreasuretro) London: Methuen, 1949. Henrici, P. [Applied and Computational Complex Analysis, Vol. 2: Special Functions, Integral Transforms, Asymptotics, Continued Fractions.](http://www.amazon.com/exec/obidos/ASIN/047154289X/ref=nosim/ericstreasuretro) New York: Wiley, pp. 322-350, 1991. Krantz, S. G. "The Laplace Transform." §15.3 in [Handbook of Complex Variables.](http://www.amazon.com/exec/obidos/ASIN/0817640118/ref=nosim/ericstreasuretro) Boston, MA: Birkhäuser, pp. 212-214, 1999. Morse, P. M. and Feshbach, H. [Methods of Theoretical Physics, Part I.](http://www.amazon.com/exec/obidos/ASIN/007043316X/ref=nosim/ericstreasuretro) New York: McGraw-Hill, pp. 467-469, 1953. Oberhettinger, F. [Tables of Laplace Transforms.](http://www.amazon.com/exec/obidos/ASIN/0387063501/ref=nosim/ericstreasuretro) New York: Springer-Verlag, 1973. Oppenheim, A. V.; Willsky, A. S.; and Nawab, S. H. [Signals and Systems, 2nd ed.](http://www.amazon.com/exec/obidos/ASIN/0138147574/ref=nosim/ericstreasuretro) Upper Saddle River, NJ: Prentice-Hall, 1997. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. [Integrals and Series, Vol. 4: Direct Laplace Transforms.](http://www.amazon.com/exec/obidos/ASIN/2881248373/ref=nosim/ericstreasuretro) New York: Gordon and Breach, 1992. Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. [Integrals and Series, Vol. 5: Inverse Laplace Transforms.](http://www.amazon.com/exec/obidos/ASIN/2881248381/ref=nosim/ericstreasuretro) New York: Gordon and Breach, 1992. Spiegel, M. R. [Theory and Problems of Laplace Transforms.](http://www.amazon.com/exec/obidos/ASIN/007060231X/ref=nosim/ericstreasuretro) New York: McGraw-Hill, 1965. Weisstein, E. W. "Books about Laplace Transforms." <http://www.ericweisstein.com/encyclopedias/books/LaplaceTransforms.html>. Widder, D. V. [The Laplace Transform.](http://www.amazon.com/exec/obidos/ASIN/0691079927/ref=nosim/ericstreasuretro) Princeton, NJ: Princeton University Press, 1941. Zwillinger, D. (Ed.). [CRC Standard Mathematical Tables and Formulae.](http://www.amazon.com/exec/obidos/ASIN/1584882913/ref=nosim/ericstreasuretro) Boca Raton, FL: CRC Press, pp. 231 and 543, 1995. ## Referenced on Wolfram\\|Alpha [Laplace Transform](https://www.wolframalpha.com/input/?i=laplace+transform "Laplace Transform") ## Cite this as: [Weisstein, Eric W.](https://mathworld.wolfram.com/about/author.html) "Laplace Transform." From [MathWorld](https://mathworld.wolfram.com/)\--A Wolfram Resource. <https://mathworld.wolfram.com/LaplaceTransform.html> ## Subject classifications - [Calculus and Analysis](https://mathworld.wolfram.com/topics/CalculusandAnalysis.html) - [Integral Transforms](https://mathworld.wolfram.com/topics/IntegralTransforms.html) - [General Integral Transforms](https://mathworld.wolfram.com/topics/GeneralIntegralTransforms.html) - [History and Terminology](https://mathworld.wolfram.com/topics/HistoryandTerminology.html) - [Wolfram Language Commands](https://mathworld.wolfram.com/topics/WolframLanguageCommands.html) - [About MathWorld](https://mathworld.wolfram.com/about/) - [MathWorld Classroom](https://mathworld.wolfram.com/classroom/) - [Contribute](https://mathworld.wolfram.com/contact/) - [MathWorld Book](https://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20) - [wolfram.com](https://www.wolfram.com/) - [13,311 Entries](https://mathworld.wolfram.com/whatsnew/) - [Last Updated: Wed Mar 25 2026](https://mathworld.wolfram.com/whatsnew/) - [©1999–2026 Wolfram Research, Inc.](https://www.wolfram.com/) - [Terms of Use](https://www.wolfram.com/legal/terms/mathworld.html) - [![Wolfram](https://mathworld.wolfram.com/images/footer/wolfram-logo.png)](https://www.wolfram.com/) - [wolfram.com](https://www.wolfram.com/) - [Wolfram for Education](https://www.wolfram.com/education/) - Created, developed and nurtured by Eric Weisstein at Wolfram Research Created, developed and nurtured by Eric Weisstein at Wolfram Research [Find out if you already have access to Wolfram tech through your organization](https://www.wolfram.com/siteinfo/) ×
Readable Markdown	[Algebra](https://mathworld.wolfram.com/topics/Algebra.html) [Applied Mathematics](https://mathworld.wolfram.com/topics/AppliedMathematics.html) [Calculus and Analysis](https://mathworld.wolfram.com/topics/CalculusandAnalysis.html) [Discrete Mathematics](https://mathworld.wolfram.com/topics/DiscreteMathematics.html) [Foundations of Mathematics](https://mathworld.wolfram.com/topics/FoundationsofMathematics.html) [Geometry](https://mathworld.wolfram.com/topics/Geometry.html) [History and Terminology](https://mathworld.wolfram.com/topics/HistoryandTerminology.html) [Number Theory](https://mathworld.wolfram.com/topics/NumberTheory.html) [Probability and Statistics](https://mathworld.wolfram.com/topics/ProbabilityandStatistics.html) [Recreational Mathematics](https://mathworld.wolfram.com/topics/RecreationalMathematics.html) [Topology](https://mathworld.wolfram.com/topics/Topology.html) [Alphabetical Index](https://mathworld.wolfram.com/letters/) [New in MathWorld](https://mathworld.wolfram.com/whatsnew/) *** The Laplace transform is an [integral transform](https://mathworld.wolfram.com/IntegralTransform.html) perhaps second only to the [Fourier transform](https://mathworld.wolfram.com/FourierTransform.html) in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear [ordinary differential equations](https://mathworld.wolfram.com/OrdinaryDifferentialEquation.html) such as those arising in the analysis of electronic circuits. The (unilateral) Laplace transform ![L](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline1.svg) (not to be confused with the [Lie derivative](https://mathworld.wolfram.com/LieDerivative.html), also commonly denoted ![L](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline2.svg)) is defined by \| \| \| \|---\|---\| \| ![ L\_t\[f(t)\](s)=int\_0^inftyf(t)e^(-st)dt, ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation1.svg) \| (1) \| where ![f(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline3.svg) is defined for ![t\>=0](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline4.svg) (Abramowitz and Stegun 1972). The unilateral Laplace transform is almost always what is meant by "the" Laplace transform, although a [bilateral Laplace transform](https://mathworld.wolfram.com/BilateralLaplaceTransform.html) is sometimes also defined as \| \| \| \|---\|---\| \| ![ L\_t^((2))\[f(t)\](s)=int\_(-infty)^inftyf(t)e^(-st)dt ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation2.svg) \| (2) \| (Oppenheim et al. 1997\). The unilateral Laplace transform ![L\_t\[f(t)\](s)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline5.svg) is implemented in the [Wolfram Language](http://www.wolfram.com/language/) as [LaplaceTransform](http://reference.wolfram.com/language/ref/LaplaceTransform.html)\[f\[t\], t, s\] and the inverse Laplace transform as [InverseRadonTransform](http://reference.wolfram.com/language/ref/InverseRadonTransform.html). The inverse Laplace transform is known as the [Bromwich integral](https://mathworld.wolfram.com/BromwichIntegral.html), sometimes known as the Fourier-Mellin integral (see also the related Duhamel's convolution principle). A table of several important one-sided Laplace transforms is given below. \| \| \| \| \|---\|---\|---\| \| ![f](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline6.svg) \| ![L\_t\[f(t)\](s)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline7.svg) \| conditions \| \| 1 \| ![1/s](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline8.svg) \| \| \| ![t](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline9.svg) \| ![1/(s^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline10.svg) \| \| \| ![t^n](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline11.svg) \| ![(n!)/(s^(n+1))](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline12.svg) \| ![n in Z\>=0](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline13.svg) \| \| ![t^a](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline14.svg) \| ![(Gamma(a+1))/(s^(a+1))](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline15.svg) \| ![R\[a\]\>-1](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline16.svg) \| \| ![e^(at)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline17.svg) \| ![1/(s-a)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline18.svg) \| \| \| ![cos(omegat)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline19.svg) \| ![s/(s^2+omega^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline20.svg) \| ![omega in R](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline21.svg) \| \| ![sin(omegat)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline22.svg) \| ![omega/(s^2+omega^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline23.svg) \| ![s\>\\|I\[omega\]\\|](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline24.svg) \| \| ![cosh(omegat)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline25.svg) \| ![s/(s^2-omega^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline26.svg) \| ![s\>\\|R\[omega\]\\|](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline27.svg) \| \| ![sinh(omegat)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline28.svg) \| ![omega/(s^2-omega^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline29.svg) \| ![s\>\\|I\[omega\]\\|](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline30.svg) \| \| ![e^(at)sin(bt)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline31.svg) \| ![b/((s-a)^2+b^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline32.svg) \| ![s\>a+\\|I\[b\]\\|](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline33.svg) \| \| ![e^(at)cos(bt)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline34.svg) \| ![(s-a)/((s-a)^2+b^2)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline35.svg) \| ![b in R](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline36.svg) \| \| ![delta(t-c)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline37.svg) \| ![e^(-cs)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline38.svg) \| \| \| ![H\_c(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline39.svg) \| ![{1/s for c\<=0; (e^(-cs))/s for c\>0](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline40.svg) \| \| \| ![J\_0(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline41.svg) \| ![1/(sqrt(s^2+1))](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline42.svg) \| \| \| ![J\_n(at)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline43.svg) \| ![((sqrt(s^2+a^2)-s)^n)/(a^nsqrt(s^2+a^2))](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline44.svg) \| ![n in Z\>=0](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline45.svg) \| In the above table, ![J\_0(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline46.svg) is the zeroth-order [Bessel function of the first kind](https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html), ![delta(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline47.svg) is the [delta function](https://mathworld.wolfram.com/DeltaFunction.html), and ![H\_c(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline48.svg) is the [Heaviside step function](https://mathworld.wolfram.com/HeavisideStepFunction.html). The Laplace transform has many important properties. The Laplace transform existence theorem states that, if ![f(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline49.svg) is [piecewise continuous](https://mathworld.wolfram.com/PiecewiseContinuous.html) on every finite interval in ![\[0,infty)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline50.svg) satisfying \| \| \| \|---\|---\| \| ![ \\|f(t)\\|\<=Me^(at) ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation3.svg) \| (3) \| for all ![t in \[0,infty)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline51.svg), then ![L\_t\[f(t)\](s)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline52.svg) exists for all ![s\>a](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline53.svg). The Laplace transform is also [unique](https://mathworld.wolfram.com/Unique.html), in the sense that, given two functions ![F\_1(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline54.svg) and ![F\_2(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline55.svg) with the same transform so that \| \| \| \|---\|---\| \| ![ L\_t\[F\_1(t)\](s)=L\_t\[F\_2(t)\](s)=f(s), ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation4.svg) \| (4) \| then [Lerch's theorem](https://mathworld.wolfram.com/LerchsTheorem.html) guarantees that the integral \| \| \| \|---\|---\| \| ![ int\_0^aN(t)dt=0 ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation5.svg) \| (5) \| vanishes for all ![a\>0](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline56.svg) for a [null function](https://mathworld.wolfram.com/NullFunction.html) defined by \| \| \| \|---\|---\| \| ![ N(t)=F\_1(t)-F\_2(t). ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation6.svg) \| (6) \| The Laplace transform is [linear](https://mathworld.wolfram.com/LinearOperator.html) since \| \| \| \| \| \|---\|---\|---\|---\| \| ![L\_t\[af(t)+bg(t)\]](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline57.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline58.svg) \| ![int\_0^infty\[af(t)+bg(t)\]e^(-st)dt](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline59.svg) \| (7) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline60.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline61.svg) \| ![aint\_0^inftyfe^(-st)dt+bint\_0^inftyge^(-st)dt](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline62.svg) \| (8) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline63.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline64.svg) \| ![aL\_t\[f(t)\]+bL\_t\[g(t)\].](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline65.svg) \| (9) \| The Laplace transform of a [convolution](https://mathworld.wolfram.com/Convolution.html) is given by \| \| \| \|---\|---\| \| ![ L\_t\[f(t)\g(t)\]=L\_t\[f(t)\]L\_t\[g(t)\] L\_t^(-1)\[FG\]=L\_t^(-1)\[F\]\L\_t^(-1)\[G\]. ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation7.svg) \| (10) \| Now consider [differentiation](https://mathworld.wolfram.com/Differentiation.html). Let ![f(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline66.svg) be continuously differentiable ![n-1](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline67.svg) times in ![\[0,infty)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline68.svg). If ![\\|f(t)\\|\<=Me^(at)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline69.svg), then \| \| \| \|---\|---\| \| ![ L\_t\[f^((n))(t)\](s)=s^nL\_t\[f(t)\]-s^(n-1)f(0)-s^(n-2)f^'(0)-...-f^((n-1))(0). ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation8.svg) \| (11) \| This can be proved by [integration by parts](https://mathworld.wolfram.com/IntegrationbyParts.html), \| \| \| \| \| \|---\|---\|---\|---\| \| ![L\_t\[f^'(t)\](s)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline70.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline71.svg) \| ![lim\_(a-\>infty)int\_0^ae^(-st)f^'(t)dt](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline72.svg) \| (12) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline73.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline74.svg) \| ![lim\_(a-\>infty){\[e^(-st)f(t)\]\_0^a+sint\_0^ae^(-st)f(t)dt}](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline75.svg) \| (13) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline76.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline77.svg) \| ![lim\_(a-\>infty)\[e^(-sa)f(a)-f(0)+sint\_0^ae^(-st)f(t)dt\]](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline78.svg) \| (14) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline79.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline80.svg) \| ![sL\_t\[f(t)\]-f(0).](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline81.svg) \| (15) \| Continuing for higher-order derivatives then gives \| \| \| \|---\|---\| \| ![ L\_t\[f^('')(t)\](s)=s^2L\_t\[f(t)\](s)-sf(0)-f^'(0). ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation9.svg) \| (16) \| This property can be used to transform differential equations into algebraic equations, a procedure known as the [Heaviside calculus](https://mathworld.wolfram.com/HeavisideCalculus.html), which can then be inverse transformed to obtain the solution. For example, applying the Laplace transform to the equation \| \| \| \|---\|---\| \| ![ f^('')(t)+a\_1f^'(t)+a\_0f(t)=0 ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation10.svg) \| (17) \| gives \| \| \| \|---\|---\| \| ![ {s^2L\_t\[f(t)\](s)-sf(0)-f^'(0)}+a\_1{sL\_t\[f(t)\](s)-f(0)} +a\_0L\_t\[f(t)\](s)=0 ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation11.svg) \| (18) \| \| \| \| \|---\|---\| \| ![ L\_t\[f(t)\](s)(s^2+a\_1s+a\_0)-sf(0)-f^'(0)-a\_1f(0)=0, ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation12.svg) \| (19) \| which can be rearranged to \| \| \| \|---\|---\| \| ![ L\_t\[f(t)\](s)=(sf(0)+f^'(0)+a\_1f(0))/(s^2+a\_1s+a\_0). ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation13.svg) \| (20) \| If this equation can be inverse Laplace transformed, then the original differential equation is solved. The Laplace transform satisfied a number of useful properties. Consider [exponentiation](https://mathworld.wolfram.com/ExponentialFunction.html). If ![L\_t\[f(t)\](s)=F(s)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline82.svg) for ![s\>alpha](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline83.svg) (i.e., ![F(s)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline84.svg) is the Laplace transform of ![f](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline85.svg)), then ![L\_t\[e^(at)f\](s)=F(s-a)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline86.svg) for ![s\>a+alpha](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline87.svg). This follows from \| \| \| \| \| \|---\|---\|---\|---\| \| ![F(s-a)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline88.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline89.svg) \| ![int\_0^inftyfe^(-(s-a)t)dt](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline90.svg) \| (21) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline91.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline92.svg) \| ![int\_0^infty\[f(t)e^(at)\]e^(-st)dt](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline93.svg) \| (22) \| \| ![](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline94.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline95.svg) \| ![L\_t\[e^(at)f(t)\](s).](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline96.svg) \| (23) \| The Laplace transform also has nice properties when applied to [integrals](https://mathworld.wolfram.com/Integral.html) of functions. If ![f(t)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline97.svg) is [piecewise continuous](https://mathworld.wolfram.com/PiecewiseContinuous.html) and ![\\|f(t)\\|\<=Me^(at)](https://mathworld.wolfram.com/images/equations/LaplaceTransform/Inline98.svg), then \| \| \| \|---\|---\| \| ![ L\_t\[int\_0^tf(t^')dt^'\]=1/sL\_t\[f(t)\](s). ](https://mathworld.wolfram.com/images/equations/LaplaceTransform/NumberedEquation14.svg) \| (24) \| *** ## See also [Bilateral Laplace Transform](https://mathworld.wolfram.com/BilateralLaplaceTransform.html), [Bromwich Integral](https://mathworld.wolfram.com/BromwichIntegral.html), [Fourier-Mellin Integral](https://mathworld.wolfram.com/Fourier-MellinIntegral.html), [Fourier Transform](https://mathworld.wolfram.com/FourierTransform.html), [Integral Transform](https://mathworld.wolfram.com/IntegralTransform.html), [Laplace-Stieltjes Transform](https://mathworld.wolfram.com/Laplace-StieltjesTransform.html), [Operational Mathematics](https://mathworld.wolfram.com/OperationalMathematics.html), [Unilateral Laplace Transform](https://mathworld.wolfram.com/UnilateralLaplaceTransform.html) [Explore this topic in the MathWorld classroom](https://mathworld.wolfram.com/classroom/LaplaceTransform.html) ## Explore with Wolfram\\|Alpha ## References Abramowitz, M. and Stegun, I. A. (Eds.). "Laplace Transforms." Ch. 29 in [Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.](http://www.amazon.com/exec/obidos/ASIN/0486612724/ref=nosim/ericstreasuretro) New York: Dover, pp. 1019-1030, 1972.Arfken, G. [Mathematical Methods for Physicists, 3rd ed.](http://www.amazon.com/exec/obidos/ASIN/0120598760/ref=nosim/ericstreasuretro) Orlando, FL: Academic Press, pp. 824-863, 1985.Churchill, R. V. [Operational Mathematics.](http://www.amazon.com/exec/obidos/ASIN/0070108706/ref=nosim/ericstreasuretro) New York: McGraw-Hill, 1958.Doetsch, G. [Introduction to the Theory and Application of the Laplace Transformation.](http://www.amazon.com/exec/obidos/ASIN/0387064079/ref=nosim/ericstreasuretro) Berlin: Springer-Verlag, 1974.Franklin, P. [An Introduction to Fourier Methods and the Laplace Transformation.](http://www.amazon.com/exec/obidos/ASIN/B0007HQ89I/ref=nosim/ericstreasuretro) New York: Dover, 1958.Graf, U. [Applied Laplace Transforms and z\-Transforms for Scientists and Engineers: A Computational Approach using a Mathematica Package.](http://www.amazon.com/exec/obidos/ASIN/3764324279/ref=nosim/weisstein-20) Basel, Switzerland: Birkhäuser, 2004.Jaeger, J. C. and Newstead, G. H. [An Introduction to the Laplace Transformation with Engineering Applications.](http://www.amazon.com/exec/obidos/ASIN/041612870X/ref=nosim/ericstreasuretro) London: Methuen, 1949.Henrici, P. [Applied and Computational Complex Analysis, Vol. 2: Special Functions, Integral Transforms, Asymptotics, Continued Fractions.](http://www.amazon.com/exec/obidos/ASIN/047154289X/ref=nosim/ericstreasuretro) New York: Wiley, pp. 322-350, 1991.Krantz, S. G. "The Laplace Transform." §15.3 in [Handbook of Complex Variables.](http://www.amazon.com/exec/obidos/ASIN/0817640118/ref=nosim/ericstreasuretro) Boston, MA: Birkhäuser, pp. 212-214, 1999.Morse, P. M. and Feshbach, H. [Methods of Theoretical Physics, Part I.](http://www.amazon.com/exec/obidos/ASIN/007043316X/ref=nosim/ericstreasuretro) New York: McGraw-Hill, pp. 467-469, 1953.Oberhettinger, F. [Tables of Laplace Transforms.](http://www.amazon.com/exec/obidos/ASIN/0387063501/ref=nosim/ericstreasuretro) New York: Springer-Verlag, 1973.Oppenheim, A. V.; Willsky, A. S.; and Nawab, S. H. [Signals and Systems, 2nd ed.](http://www.amazon.com/exec/obidos/ASIN/0138147574/ref=nosim/ericstreasuretro) Upper Saddle River, NJ: Prentice-Hall, 1997.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. [Integrals and Series, Vol. 4: Direct Laplace Transforms.](http://www.amazon.com/exec/obidos/ASIN/2881248373/ref=nosim/ericstreasuretro) New York: Gordon and Breach, 1992.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. [Integrals and Series, Vol. 5: Inverse Laplace Transforms.](http://www.amazon.com/exec/obidos/ASIN/2881248381/ref=nosim/ericstreasuretro) New York: Gordon and Breach, 1992.Spiegel, M. R. [Theory and Problems of Laplace Transforms.](http://www.amazon.com/exec/obidos/ASIN/007060231X/ref=nosim/ericstreasuretro) New York: McGraw-Hill, 1965.Weisstein, E. W. "Books about Laplace Transforms." <http://www.ericweisstein.com/encyclopedias/books/LaplaceTransforms.html>.Widder, D. V. [The Laplace Transform.](http://www.amazon.com/exec/obidos/ASIN/0691079927/ref=nosim/ericstreasuretro) Princeton, NJ: Princeton University Press, 1941.Zwillinger, D. (Ed.). [CRC Standard Mathematical Tables and Formulae.](http://www.amazon.com/exec/obidos/ASIN/1584882913/ref=nosim/ericstreasuretro) Boca Raton, FL: CRC Press, pp. 231 and 543, 1995. ## Referenced on Wolfram\\|Alpha [Laplace Transform](https://www.wolframalpha.com/input/?i=laplace+transform "Laplace Transform") ## Cite this as: [Weisstein, Eric W.](https://mathworld.wolfram.com/about/author.html) "Laplace Transform." From [MathWorld](https://mathworld.wolfram.com/)\--A Wolfram Resource. <https://mathworld.wolfram.com/LaplaceTransform.html> ## Subject classifications
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