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curl -X POST \
  'http://laksa120.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://mathvault.ca/laplace-transform/\')), getAhrefsUnparsedNoserviceFromURL(\'https://mathvault.ca/laplace-transform/\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/mathvault.ca\/laplace-transform\/","crawl_time":1776425059,"first_indexed_time":1596973748,"http_code":200,"src_unparsed":"ca,mathvault!\/laplace-transform\/ s443","src_root_hash":"8577111680796856520","history_drop_reason":null,"meta_title":"Laplace Transform: A First Introduction | Math Vault","meta_descriptions":["A gentle, concise introduction to the concept of Laplace transform, along with 9 basic examples to illustrate its derivations and usage."],"attrs_boilerpipe_text":"Let us take a moment to ponder how truly bizarre the\nLaplace transform\nis.\nYou put in a sine and get an oddly simple,\narbitrary-looking fraction\n. Why do we suddenly have squares?\nYou look at the table of\ncommon Laplace transforms\nto find a pattern and you see no rhyme, no reason, no obvious link between different functions and their different, very different, results.\nWhat’s going on here?\nOr so we thought when we first encountered the cursive\nL\nin school.\nTable of Contents\nWhat does the Laplace transform do, really?\nSome Preliminary Examples\nLooking Inside the Laplace Transform of Sine\nDiverging Functions: What the Laplace Transform is for\nA Transform of Unfathomable Power\nWhat does the Laplace transform do, really?\nAt a high level, Laplace transform is an\nintegral transform\nmostly encountered in differential equations — in electrical engineering for instance — where electric circuits are represented as differential equations.\nIn fact, it takes a\ntime-domain function\n, where\nt\nis the variable, and outputs a\nfrequency-domain function\n, where\ns\nis the variable. Definition-wise, Laplace transform takes a function of real variable\nf\n(\nt\n)\n(defined for all\nt\n≥\n0\n) to a function of complex variable\nF\n(\ns\n)\nas follows:\nL\n{\nf\n(\nt\n)\n}\n=\n∫\n0\n∞\nf\n(\nt\n)\ne\n−\ns\nt\nd\nt\n=\nF\n(\ns\n)\nSome Preliminary Examples\nWhat fate awaits\nsimple functions\nas they enter the Laplace transform?\nTake the simplest function: the\nconstant function\nf\n(\nt\n)\n=\n1\n. In this case, putting\n1\nin the transform yields\n1\n\/\ns\n, which means that we went from a constant to a variable-dependent function.\n(Odd but not too worrying. After all, we’ve seen\n1\n\/\nx\nintegrating to\nln\n⁡\nx\nin calculus. Not a constant-to-variable situation of course, but an unexpected transformation nonetheless.)\nLet us take it up a notch, with the\nlinear function\nf\n(\nt\n)\n=\nt\n. After the transformation, it is turned into\n1\n\/\ns\n2\n, which means that we went from\n1\n→\n1\n\/\ns\nto\nt\n→\n1\n\/\ns\n2\n. A pattern begins to emerge.\nNow what about\nf\n(\nt\n)\n=\nt\nn\n? With this simple\npower function\n, we end up with:\nL\n{\nt\nn\n}\n=\nn\n!\ns\nn\n+\n1\nSo there was a factorial in\nL\n{\nt\n}\nall along, hidden by the fact that\n1\n!\n=\n1\n. What else is the transform hiding?\nHere, a glance at a table of\ncommon Laplace transforms\nwould show that the emerging pattern cannot explain other functions easily. Things get weird, and the weirdness escalates quickly — which brings us back to the sine function.\nLooking Inside the Laplace Transform of Sine\nLet us unpack what happens to our sine function as we Laplace-transform it. We begin by noticing that a sine function can be expressed as a\ncomplex exponential\n— an indirect result of the celebrated\nEuler’s formula\n:\ne\ni\nt\n=\ncos\n⁡\nt\n+\ni\nsin\n⁡\nt\nIn fact, a sine is often expressed in terms of exponentials for\nease of calculation\n, so if we apply that to the function\nf\n(\nt\n)\n=\nsin\n⁡\n(\na\nt\n)\n, we would get:\nsin\n⁡\n(\na\nt\n)\n=\ne\ni\na\nt\n−\ne\n−\ni\na\nt\n2\ni\nThus the\nLaplace transform\nof\nsin\n⁡\n(\na\nt\n)\nthen becomes:\nL\n{\nsin\n⁡\n(\na\nt\n)\n}\n=\n1\n2\ni\n∫\n0\n∞\n(\ne\ni\na\nt\n−\ne\n−\ni\na\nt\n)\ne\n−\ns\nt\nd\nt\nwhich means that we have a\nproduct of exponentials\n. Distributing the terms, we get:\nL\n{\nsin\n⁡\n(\na\nt\n)\n}\n=\n1\n2\ni\n∫\n0\n∞\ne\ni\na\nt\n−\ns\nt\n−\ne\n−\ni\na\nt\n−\ns\nt\nd\nt\nHere,\nfactoring\nthe\nt\nin the exponents yields:\nL\n{\nsin\n⁡\n(\na\nt\n)\n}\n=\n1\n2\ni\n∫\n0\n∞\ne\n(\ni\na\n−\ns\n)\nt\n−\ne\n(\n−\ni\na\n−\ns\n)\nt\nd\nt\nand since\nRe\n(\ns\n)\n>\n0\nby assumption, we can proceed with the integration from\n0\nto\n∞\nas usual:\nL\n{\nsin\n⁡\n(\na\nt\n)\n}\n=\ne\n(\ni\na\n−\ns\n)\nt\n2\ni\n(\ni\na\n−\ns\n)\n|\n0\n∞\n−\ne\n(\n−\ni\na\n−\ns\n)\nt\n2\ni\n(\n−\ni\na\n−\ns\n)\n|\n0\n∞\nLet us simplify further.\nDistributing\nthe\ni\ninside the parentheses, we get:\nL\n{\nsin\n⁡\n(\na\nt\n)\n}\n=\ne\n(\ni\na\n−\ns\n)\nt\n2\n(\n−\na\n−\ni\ns\n)\n|\n0\n∞\n−\ne\n(\n−\ni\na\n−\ns\n)\nt\n2\n(\na\n−\ni\ns\n)\n|\n0\n∞\nBy evaluating the\nt\nat the\nboundaries\n, we get:\nL\n{\nsin\n⁡\n(\na\nt\n)\n}\n=\n(\ne\n(\ni\na\n−\ns\n)\n⋅\n∞\n2\n(\n−\na\n−\ni\ns\n)\n−\ne\n(\ni\na\n−\ns\n)\n⋅\n0\n2\n(\n−\na\n−\ni\ns\n)\n)\n−\n(\ne\n(\n−\ni\na\n−\ns\n)\n⋅\n∞\n2\n(\na\n−\ni\ns\n)\n−\ne\n(\n−\ni\na\n−\ns\n)\n⋅\n0\n2\n(\na\n−\ni\ns\n)\n)\nAnd because\nRe\n(\ns\n)\n>\n0\nby assumption, both\ne\n(\ni\na\n−\ns\n)\n⋅\n∞\nand\ne\n(\n−\ni\na\n−\ns\n)\n⋅\n∞\noscillate to\n0\n(i.e.,\nvanish at infinity\n), after which we are then left with:\nL\n{\nsin\n⁡\n(\na\nt\n)\n}\n=\n1\n−\n2\n(\n−\na\n−\ni\ns\n)\n+\n1\n2\n(\na\n−\ni\ns\n)\nOnce there, merging the\nfractions\ntogether would yield:\nL\n{\nsin\n⁡\n(\na\nt\n)\n}\n=\n2\n(\na\n−\ni\ns\n)\n−\n2\n(\n−\na\n−\ni\ns\n)\n−\n4\n(\na\n−\ni\ns\n)\n(\n−\na\n−\ni\ns\n)\n=\n2\na\n−\n2\ni\ns\n+\n2\na\n+\n2\ni\ns\n4\n(\na\n2\n+\ni\ns\na\n−\ni\ns\na\n+\ns\n2\n)\n=\n4\na\n4\n(\na\n2\n+\ns\n2\n)\n=\na\na\n2\n+\ns\n2\nwhich shows that after Laplace transform, a sine is turned into a more tractable\ngeometric function\n. By following similar reasoning, the Laplace transform of cosine can be shown to be equal to the following expression as well:\nL\n{\ncos\n⁡\n(\na\nt\n)\n}\n=\ns\na\n2\n+\ns\n2\n(\nRe\n(\ns\n)\n>\n0\n)\nBut then, one might argue “Why do we need to transform trigonometric functions like this when we can just\nintegrate\nthem?”\nDiverging Functions: What the Laplace Transform is for\nWhat if we throw a wrench in there by introducing a\ndiverging function\n, say,\nf\n(\nt\n)\n=\ne\na\nt\n? As it turns out, the Laplace transform of the exponential\ne\na\nt\nis actually deceptively simple:\nL\n{\ne\na\nt\n}\n=\n∫\n0\n∞\ne\na\nt\ne\n−\ns\nt\nd\nt\n=\n∫\n0\n∞\ne\n(\na\n−\ns\n)\nt\nd\nt\nHere, we see that so long as\nRe\n(\ns\n)\n>\na\n, we would get that:\n∫\n0\n∞\ne\n(\na\n−\ns\n)\nt\nd\nt\n=\ne\n(\na\n−\ns\n)\nt\na\n−\ns\n|\n0\n∞\n=\n0\n−\n1\na\n−\ns\n=\n1\ns\n−\na\nThat is, as long as\nRe\n(\ns\n)\n>\na\n, the Laplace transform of\ne\na\nt\nis a simple\n1\n\/\n(\ns\n−\na\n)\n. Here’s a\nvideo version\nof the derivation for the record.\nOn the other hand, if we mix the exponential\ne\na\nt\nwith the\npower function\nt\nn\n, we would then have:\nL\n{\nt\nn\ne\na\nt\n}\n=\n∫\n0\n∞\nt\nn\ne\na\nt\ne\n−\ns\nt\nd\nt\nwhich, after a bit of\nrecursion\nand\nintegration by parts\n, would become:\nn\n!\n(\ns\n−\na\n)\nn\n+\n1\nHere, notice how the transforms of exponential and power function are both\nrepresented\nin the expression, with the factorial\nn\n!\n, the\n1\n\/\n(\ns\n−\na\n)\nfraction, and the\nn\n+\n1\nexponent.\nIn fact, it turns out that we can integrate\nany\nfunction with the Laplace transform, as long as it does not\ndiverge\nfaster than the\ne\na\nt\nexponential. In the tables of Laplace transforms, you might have noticed the\nRe\n(\ns\n)\n>\na\ncondition. That is what the condition is alluding to.\nA Transform of Unfathomable Power\nHowever, what we have seen is only the tip of the iceberg, since we can also use Laplace transform to transform the\nderivatives\nas well. In goes\nf\n(\nn\n)\n(\nt\n)\n. Something happens. Then out goes:\ns\nn\nL\n{\nf\n(\nt\n)\n}\n−\n∑\nr\n=\n0\nn\n−\n1\ns\nn\n−\n1\n−\nr\nf\n(\nr\n)\n(\n0\n)\nFor example, when\nn\n=\n2\n, we have that:\nL\n{\nf\n′\n′\n(\nt\n)\n}\n=\ns\n2\nL\n{\nf\n(\nt\n)\n}\n−\ns\nf\n(\n0\n)\n−\nf\n′\n(\n0\n)\nIn addition to the derivatives, the\nL\ncan also process some\nintegrals\n: the integral sine, cosine and exponential, as well as the\nerror function\n— to name a few.\nBut that’s not all. There is also the\ninverse Laplace transform\n, which takes a frequency-domain function and renders a time-domain function.\nIn fact, performing the transform from time to frequency and back once introduces a factor of\n1\n\/\n2\nπ\n. Sometimes, you’ll see the whole fraction in front of the inverse function, while other times, the transform and its inverse share a factor of\n1\n\/\n2\nπ\n.\nThis is as if the Kraken could restitute the boat intact — but only for a factor of\n1\n\/\n2\nπ\n.\nThe Laplace transform, even after all those years, never ceases to bring us awe with its power. Here’s a\ntable\nsummarizing the transforms we’ve discussed thus far:\nFunction\nLaplace Transform\n1\n1\ns\nt\n1\ns\n2\nt\nn\nn\n!\ns\nn\n+\n1\ne\na\nt\n1\ns\n−\na\nsin\n⁡\n(\na\nt\n)\na\na\n2\n+\ns\n2\ncos\n⁡\n(\na\nt\n)\ns\na\n2\n+\ns\n2\nt\nn\ne\na\nt\nn\n!\n(\ns\n−\na\n)\nn\n+\n1\nf\n(\n2\n)\n(\nt\n)\ns\n2\nL\n{\nf\n(\nt\n)\n}\n−\ns\nf\n(\n0\n)\n−\nf\n′\n(\n0\n)\nf\n(\nn\n)\n(\nt\n)\ns\nn\nL\n{\nf\n(\nt\n)\n}\n−\n∑\nr\n=\n0\nn\n−\n1\ns\nn\n−\n1\n−\nr\nf\n(\nr\n)\n(\n0\n)","attrs_markdown":"![](https:\/\/www.facebook.com\/tr?id=3257118080998043&ev=PageView&noscript=1)\n\n[![Light logo of Math Vault, an online resource hub of higher mathematics](https:\/\/mathvault.ca\/wp-content\/uploads\/New-MVLogo-White.png)](https:\/\/mathvault.ca\/)\n\n- [Learn](https:\/\/mathvault.ca\/laplace-transform\/)\n  - [Hub](https:\/\/mathvault.ca\/hub\/)\n    Definitive resource hub on everything higher math\n  - [Vault](https:\/\/mathvault.ca\/vault\/)\n    Bonus guides and lessons on mathematics and other related topics\n- [Products & Services](https:\/\/mathvault.ca\/shop)\n  - [Definitive Guide to Learning Higher Mathematics](https:\/\/mathvault.ca\/higher-math-learning-guide\/)\n  - [Ultimate LaTeX Reference Guide](https:\/\/mathvault.ca\/latex\/)\n  - [Comprehensive List of Mathematical Symbols](https:\/\/mathvault.ca\/product\/comprehensive-list-of-mathematical-symbols\/)\n  - [Math Tutoring \/ Consulting](https:\/\/mathvault.ca\/tutoring\/)\n- [Resources]()\n  - [Higher Math Proficiency Test](https:\/\/mathvault.ca\/math-test\/)\n  - [10 Commandments of Higher Math Learning](https:\/\/mathvault.ca\/10-commandments\/)\n  - [Math Vault Linear Algebra Ebook Series](https:\/\/mathvault.ca\/free-ebooks-series\/)\n  - [Recommended Math Books](https:\/\/mathvault.ca\/books\/)\n  - [Recommended Math Websites](https:\/\/mathvault.ca\/websites\/)\n  - [Recommended Online Tools](https:\/\/mathvault.ca\/online-tools\/)\n- [About]()\n  - [About Us](https:\/\/mathvault.ca\/about\/)\n    Where we came from, and where we're going\n  - [Write for Us](https:\/\/mathvault.ca\/write-for-us\/)\n    Join us in contributing to the glory of mathematics\n\n[Start Here](https:\/\/mathvault.ca\/introduction-series\/)\n\n[Kim Thibault](https:\/\/mathvault.ca\/author\/kimthibo\/ \"Kim Thibault\")\n\n# Laplace Transform: A First Introduction\nCollege Math, Calculus, Complex Number\n\n- [Home](https:\/\/mathvault.ca\/)\n- »\n- [Vault](https:\/\/mathvault.ca\/vault\/)\n- »\n- [College Math](https:\/\/mathvault.ca\/college-math\/)\n- »\n- Laplace Transform: A First Introduction\n\nLet us take a moment to ponder how truly bizarre the **[Laplace transform](https:\/\/mathvault.ca\/hub\/higher-math\/math-symbols\/calculus-analysis-symbols\/#Key_Transforms)** is.\n\nYou put in a sine and get an oddly simple, **arbitrary-looking fraction**. Why do we suddenly have squares?\n\nYou look at the table of **common Laplace transforms** to find a pattern and you see no rhyme, no reason, no obvious link between different functions and their different, very different, results.\n\nWhat’s going on here?\n\nOr so we thought when we first encountered the cursive L in school.\n\n![Laplace transform of function f](https:\/\/mathvault.ca\/wp-content\/uploads\/Laplace-Transform.png)\n\nTable of Contents\n\n[Toggle](https:\/\/mathvault.ca\/laplace-transform\/)\n\n- [What does the Laplace transform do, really?](https:\/\/mathvault.ca\/laplace-transform\/#What_does_the_Laplace_transform_do_really)\n- [Some Preliminary Examples](https:\/\/mathvault.ca\/laplace-transform\/#Some_Preliminary_Examples)\n- [Looking Inside the Laplace Transform of Sine](https:\/\/mathvault.ca\/laplace-transform\/#Looking_Inside_the_Laplace_Transform_of_Sine)\n- [Diverging Functions: What the Laplace Transform is for](https:\/\/mathvault.ca\/laplace-transform\/#Diverging_Functions_What_the_Laplace_Transform_is_for)\n- [A Transform of Unfathomable Power](https:\/\/mathvault.ca\/laplace-transform\/#A_Transform_of_Unfathomable_Power)\n## [What does the Laplace transform do, really?](https:\/\/mathvault.ca\/laplace-transform\/#toc)\nAt a high level, Laplace transform is an **[integral transform](https:\/\/en.wikipedia.org\/wiki\/Integral_transform#:~:text=In%20mathematics%2C%20an%20integral%20transform,in%20the%20original%20function%20space.)** mostly encountered in differential equations — in electrical engineering for instance — where electric circuits are represented as differential equations.\n\nIn fact, it takes a **time-domain function**, where t is the variable, and outputs a **frequency-domain function**, where s is the variable. Definition-wise, Laplace transform takes a function of real variable f ( t ) (defined for all t ≥ 0) to a function of complex variable F ( s ) as follows:\n\nL\n\n{\n\nf\n\n(\n\nt\n\n)\n\n}\n\n\\=\n\n∫\n\n0\n\n∞\n\nf\n\n(\n\nt\n\n)\n\ne\n\n−\n\ns\n\nt\n\nd\n\nt\n\n\\=\n\nF\n\n(\n\ns\n\n)\n## [Some Preliminary Examples](https:\/\/mathvault.ca\/laplace-transform\/#toc)\nWhat fate awaits **simple functions** as they enter the Laplace transform?\n\nTake the simplest function: the **constant function** f ( t ) \\= 1. In this case, putting 1 in the transform yields 1 \/ s, which means that we went from a constant to a variable-dependent function.\n\n(Odd but not too worrying. After all, we’ve seen 1 \/ x integrating to ln ⁡ x in calculus. Not a constant-to-variable situation of course, but an unexpected transformation nonetheless.)\n\nLet us take it up a notch, with the **linear function** f ( t ) \\= t. After the transformation, it is turned into 1 \/ s 2, which means that we went from 1 → 1 \/ s to t → 1 \/ s 2. A pattern begins to emerge.\n\nNow what about f ( t ) \\= t n? With this simple **power function**, we end up with: L { t n } \\= n \\! s n \\+ 1 So there was a factorial in L { t } all along, hidden by the fact that 1 \\! \\= 1. What else is the transform hiding?\n\nHere, a glance at a table of **[common Laplace transforms](https:\/\/tutorial.math.lamar.edu\/pdf\/Laplace_Table.pdf)** would show that the emerging pattern cannot explain other functions easily. Things get weird, and the weirdness escalates quickly — which brings us back to the sine function.\n\n## [Looking Inside the Laplace Transform of Sine](https:\/\/mathvault.ca\/laplace-transform\/#toc)\nLet us unpack what happens to our sine function as we Laplace-transform it. We begin by noticing that a sine function can be expressed as a **complex exponential** — an indirect result of the celebrated [Euler’s formula](https:\/\/mathvault.ca\/euler-formula\/):e i t \\= cos ⁡ t \\+ i sin ⁡ tIn fact, a sine is often expressed in terms of exponentials for **ease of calculation**, so if we apply that to the function f ( t ) \\= sin ⁡ ( a t ), we would get: sin ⁡ ( a t ) \\= e i a t − e − i a t 2 iThus the **Laplace transform** of sin ⁡ ( a t ) then becomes:  \nL { sin ⁡ ( a t ) } \\= 1 2 i ∫ 0 ∞ ( e i a t − e − i a t ) e − s t d twhich means that we have a **product of exponentials**. Distributing the terms, we get:  \nL { sin ⁡ ( a t ) } \\= 1 2 i ∫ 0 ∞ e i a t − s t − e − i a t − s t d t\n\nHere, **factoring** the t in the exponents yields:  \nL { sin ⁡ ( a t ) } \\= 1 2 i ∫ 0 ∞ e ( i a − s ) t − e ( − i a − s ) t d tand since Re ( s ) \\> 0 by assumption, we can proceed with the integration from 0 to ∞ as usual:  \nL { sin ⁡ ( a t ) } \\= e ( i a − s ) t 2 i ( i a − s ) \\| 0 ∞ − e ( − i a − s ) t 2 i ( − i a − s ) \\| 0 ∞\n\nLet us simplify further. **Distributing** the i inside the parentheses, we get:  \nL { sin ⁡ ( a t ) } \\= e ( i a − s ) t 2 ( − a − i s ) \\| 0 ∞ − e ( − i a − s ) t 2 ( a − i s ) \\| 0 ∞By evaluating the t at the **boundaries**, we get:  \nL { sin ⁡ ( a t ) } \\= ( e ( i a − s ) ⋅ ∞ 2 ( − a − i s ) − e ( i a − s ) ⋅ 0 2 ( − a − i s ) ) − ( e ( − i a − s ) ⋅ ∞ 2 ( a − i s ) − e ( − i a − s ) ⋅ 0 2 ( a − i s ) )And because Re ( s ) \\> 0 by assumption, both e ( i a − s ) ⋅ ∞ and e ( − i a − s ) ⋅ ∞ oscillate to 0 (i.e., **[vanish at infinity](https:\/\/mathvault.ca\/math-glossary\/#vanish)**), after which we are then left with:L { sin ⁡ ( a t ) } \\= 1 − 2 ( − a − i s ) \\+ 1 2 ( a − i s )Once there, merging the **fractions** together would yield:L { sin ⁡ ( a t ) } \\= 2 ( a − i s ) − 2 ( − a − i s ) − 4 ( a − i s ) ( − a − i s ) \\= 2 a − 2 i s \\+ 2 a \\+ 2 i s 4 ( a 2 \\+ i s a − i s a \\+ s 2 ) \\= 4 a 4 ( a 2 \\+ s 2 ) \\= a a 2 \\+ s 2which shows that after Laplace transform, a sine is turned into a more tractable **geometric function**. By following similar reasoning, the Laplace transform of cosine can be shown to be equal to the following expression as well: L { cos ⁡ ( a t ) } \\= s a 2 \\+ s 2 ( Re ( s ) \\> 0 ) But then, one might argue “Why do we need to transform trigonometric functions like this when we can just **integrate** them?”\n\n## [Diverging Functions: What the Laplace Transform is for](https:\/\/mathvault.ca\/laplace-transform\/#toc)\nWhat if we throw a wrench in there by introducing a **diverging function**, say, f ( t ) \\= e a t? As it turns out, the Laplace transform of the exponential e a t is actually deceptively simple: L { e a t } \\= ∫ 0 ∞ e a t e − s t d t \\= ∫ 0 ∞ e ( a − s ) t d tHere, we see that so long as Re ( s ) \\> a, we would get that: ∫ 0 ∞ e ( a − s ) t d t \\= e ( a − s ) t a − s \\| 0 ∞ \\= 0 − 1 a − s \\= 1 s − a That is, as long as Re ( s ) \\> a, the Laplace transform of e a t is a simple 1 \/ ( s − a ). Here’s a **video version** of the derivation for the record.\n\nOn the other hand, if we mix the exponential e a t with the **power function** t n, we would then have: L { t n e a t } \\= ∫ 0 ∞ t n e a t e − s t d t which, after a bit of [recursion](https:\/\/mathvault.ca\/math-glossary\/#recursion) and [integration by parts](https:\/\/mathvault.ca\/integration-overshooting-method\/#Integration_By_Parts), would become:n \\! ( s − a ) n \\+ 1Here, notice how the transforms of exponential and power function are both **represented** in the expression, with the factorial n \\!, the 1 \/ ( s − a ) fraction, and the n \\+ 1 exponent.\n\nIn fact, it turns out that we can integrate *any* function with the Laplace transform, as long as it does not **diverge** faster than the e a t exponential. In the tables of Laplace transforms, you might have noticed the Re ( s ) \\> a condition. That is what the condition is alluding to.\n\n## [A Transform of Unfathomable Power](https:\/\/mathvault.ca\/laplace-transform\/#toc)\nHowever, what we have seen is only the tip of the iceberg, since we can also use Laplace transform to transform the **derivatives** as well. In goes f ( n ) ( t ). Something happens. Then out goes:s n L { f ( t ) } − ∑ r \\= 0 n − 1 s n − 1 − r f ( r ) ( 0 )For example, when n \\= 2, we have that:L { f ′ ′ ( t ) } \\= s 2 L { f ( t ) } − s f ( 0 ) − f ′ ( 0 )In addition to the derivatives, the L can also process some **integrals**: the integral sine, cosine and exponential, as well as the [error function](https:\/\/mathvault.ca\/hub\/higher-math\/math-symbols\/probability-statistics-symbols\/#Continuous_Probability_Distributions_and_Associated_Functions) — to name a few.\n\nBut that’s not all. There is also the **[inverse Laplace transform](https:\/\/mathvault.ca\/hub\/higher-math\/math-symbols\/calculus-analysis-symbols\/#Key_Transforms)**, which takes a frequency-domain function and renders a time-domain function.\n\nIn fact, performing the transform from time to frequency and back once introduces a factor of 1 \/ 2 π. Sometimes, you’ll see the whole fraction in front of the inverse function, while other times, the transform and its inverse share a factor of 1 \/ 2 π.\n\nThis is as if the Kraken could restitute the boat intact — but only for a factor of 1 \/ 2 π.\n\nThe Laplace transform, even after all those years, never ceases to bring us awe with its power. Here’s a **table** summarizing the transforms we’ve discussed thus far:\n\n| Function | Laplace Transform |\n|---|---|\n| 1 | 1 s |\n| t | 1 s 2 |\n| t n | n \\! s n \\+ 1 |\n| e a t | 1 s − a |\n| sin ⁡ ( a t ) | a a 2 \\+ s 2 |\n| cos ⁡ ( a t ) | s a 2 \\+ s 2 |\n| t n e a t | n \\! ( s − a ) n \\+ 1 |\n| f ( 2 ) ( t ) | s 2 L { f ( t ) } − s f ( 0 ) − f ′ ( 0 ) |\n| f ( n ) ( t ) | s n L { f ( t ) } − ∑ r \\= 0 n − 1 s n − 1 − r f ( r ) ( 0 ) |\n\n[![The Definitive Guide to Higher Mathematics — Cover](https:\/\/mathvault.ca\/wp-content\/uploads\/Higher-Math-Guide-Cover.png)](https:\/\/mathvault.ca\/higher-math-learning-guide\/)\n\nHow's your higher math going?\n\nShallow learning and mechanical practices rarely work in higher mathematics. Instead, use these 10 principles to **optimize your learning** and prevent years of wasted effort.\n\n[Hmm... **Tell me more.**](https:\/\/mathvault.ca\/higher-math-learning-guide\/)\n\n- 100shares\n\n![Math Vault Standard Post](https:\/\/mathvault.ca\/wp-content\/litespeed\/avatar\/8f28b1715d6e47ed7ec221f47fa20fc9.jpg?ver=1775679107)\n\nKim Thibault\n\n#### About the author\nKim Thibault is an incorrigible polymath. After a Ph.D. in Physics, she did applied research in machine learning for audio, then a stint in programming, to finally become an author and scientific translator. She occasionally solves differential equations as a hobby. Find her on her [**website**](http:\/\/kimthibault.com\/) and on [**Linkedin**](https:\/\/linkedin.com\/in\/kimthibaultphd).\n\n[« Previous PostThe Definitive Glossary of Higher Mathematical Jargon](https:\/\/mathvault.ca\/math-glossary\/)\n\n[Next Post »Euler’s Formula: A Complete Guide](https:\/\/mathvault.ca\/euler-formula\/)\n#### You may also like\n[![Diagram illustrating Euler's formula for complex numbers](https:\/\/mathvault.ca\/wp-content\/uploads\/Euler-formula-diagram.png)](https:\/\/mathvault.ca\/euler-formula\/ \"Euler’s Formula: A Complete Guide\")\n\nEuler’s Formula: A Complete Guide\n\n[Euler’s Formula: A Complete Guide](https:\/\/mathvault.ca\/euler-formula\/)\n\n[![Header image of Math Vault's A Primer on Statistical Significance](https:\/\/mathvault.ca\/wp-content\/uploads\/Statistical-Significance.jpg)](https:\/\/mathvault.ca\/statistical-significance\/ \"A First Introduction to Statistical Significance — Through Dice Rolling and Other Uncanny Examples\")\n\nA First Introduction to Statistical Significance — Through Dice Rolling and Other Uncanny Examples\n\n[A First Introduction to Statistical Significance — Through Dice Rolling and Other Uncanny Examples](https:\/\/mathvault.ca\/statistical-significance\/)\n\n#### Leave a Reply\n[Cancel reply](https:\/\/mathvault.ca\/laplace-transform\/#respond)\n\n1. ![](https:\/\/mathvault.ca\/wp-content\/litespeed\/avatar\/8d683f12d5f4861c291bb898dfd8242b.jpg?ver=1775608862)\n   \n   **Precious Amadi** says:\n   \n   [February 10, 2021 at 2:11 AM](https:\/\/mathvault.ca\/laplace-transform\/#comment-3586)\n   \n   Any link to download soft copy?\n   \n   [Reply](https:\/\/mathvault.ca\/laplace-transform\/#comment-3586)\n   1. [![](https:\/\/mathvault.ca\/wp-content\/litespeed\/avatar\/d96413892cebda856b0216ca7e7f9f76.jpg?ver=1775604434)](https:\/\/mathvault.ca\/) **[Math Vault](https:\/\/mathvault.ca\/)** says:\n      \n      [February 10, 2021 at 2:02 PM](https:\/\/mathvault.ca\/laplace-transform\/#comment-3587)\n      \n      Hi Amadi. Thanks for your interest. If you click on the “Print” button in your browser, you can save the guide as a PDF. That would be a way of generating a soft copy.\n      \n      [Reply](https:\/\/mathvault.ca\/laplace-transform\/#comment-3587)\n\n{\"email\":\"Email address invalid\",\"url\":\"Website address invalid\",\"required\":\"Required field missing\"}\n\n#### Search\n***\n#### Navigational Menu\n***\n[**General** **M****a****th**](https:\/\/mathvault.ca\/general-math\/)  \n [**Alg****e****bra**](https:\/\/mathvault.ca\/general-math\/algebra\/)  \n [**Func****tions &** **Operations**](https:\/\/mathvault.ca\/general-math\/functions-operations\/)  \n[**College** **M****ath**](https:\/\/mathvault.ca\/college-math)  \n[**Calc****ul****us**](https:\/\/mathvault.ca\/college-math\/calculus\/)  \n[**Probab****ilit****y & Statistics**](https:\/\/mathvault.ca\/college-math\/probability-statistics\/)  \n[**Foundation** **o****f Higher Math**](https:\/\/mathvault.ca\/foundation-of-higher-math\/)  \n[**Math** **Too****ls**](https:\/\/mathvault.ca\/math-tools\/)\n\n#### Useful Resources\n***\n[Higher Math Exploration Series](https:\/\/mathvault.ca\/introduction-series\/)  \n[10 Commandments of Higher Math Learning](https:\/\/mathvault.ca\/10-commandments\/)  \n[Compendium of Math Symbols](https:\/\/mathvault.ca\/hub\/higher-math\/math-symbols)  \n[Higher Math Proficiency Test](https:\/\/mathvault.ca\/math-test\/)\n\n#### Math eBooks\n***\n[Definitive Guide to Learning Higher Math](https:\/\/mathvault.ca\/higher-math-learning-guide\/)  \n[Ultimate LaTeX Reference Guide](https:\/\/mathvault.ca\/latex\/)  \n[Linear Algebra eBook Series](https:\/\/mathvault.ca\/free-ebooks-series\/)\n\n#### Popular Guides\n***\n## [The Definitive Glossary of Higher Mathematical Jargon](https:\/\/mathvault.ca\/math-glossary\/ \"The Definitive Glossary of Higher Mathematical Jargon\")\n[The Definitive Glossary of Higher Mathematical Jargon](https:\/\/mathvault.ca\/math-glossary\/)\n\n## [The Definitive, Non-Technical Introduction to LaTeX, Professional Typesetting and Scientific Publishing](https:\/\/mathvault.ca\/latex-guide\/ \"The Definitive, Non-Technical Introduction to LaTeX, Professional Typesetting and Scientific Publishing\")\n[The Definitive, Non-Technical Introduction to LaTeX, Professional Typesetting and Scientific Publishing](https:\/\/mathvault.ca\/latex-guide\/)\n\n## [The Definitive Higher Math Guide on Integer Long Division (and Its Variants)](https:\/\/mathvault.ca\/long-division\/ \"The Definitive Higher Math Guide on Integer Long Division (and Its Variants)\")\n[The Definitive Higher Math Guide on Integer Long Division (and Its Variants)](https:\/\/mathvault.ca\/long-division\/)\n\nAbout Math Vault\n\nOriginally founded as a Montreal-based math tutoring agency, Math Vault has since then morphed into a global resource hub for people interested in learning more about higher mathematics. For more, see [about us](https:\/\/mathvault.ca\/about\/).\n\nGuides\n\n[Definitive Guide to Learning Higher Math](https:\/\/mathvault.ca\/higher-math-learning-guide\/)\n\n[Ultimate LaTeX Reference Guide](https:\/\/mathvault.ca\/latex\/)\n\n[Comprehensive List of Math Symbols](https:\/\/mathvault.ca\/product\/comprehensive-list-of-mathematical-symbols\/)\n\n[Linear Algebra eBook Series](https:\/\/mathvault.ca\/free-ebooks-series\/)\n\nResources & Tools\n\n[Higher Math Exploration Series](https:\/\/mathvault.ca\/introduction-series\/)\n\n[Higher Math Proficiency Test](https:\/\/mathvault.ca\/math-test\/)\n\n[10 Commandments of Higher Math Learning](https:\/\/mathvault.ca\/10-commandments\/)\n\nContact\n\n[About Us](https:\/\/mathvault.ca\/about\/)\n\n[Write for Us](https:\/\/mathvault.ca\/write-for-us\/)\n\n© 2026 Math Vault. All Rights Reserved.\n\n[Privacy Policy](https:\/\/mathvault.ca\/privacy-policy\/) [Terms of Use](https:\/\/mathvault.ca\/terms-of-use\/) [Anti-Spam](https:\/\/mathvault.ca\/anti-spam-policy\/) [Disclosure](https:\/\/mathvault.ca\/affiliate-disclosure\/) [DMCA Notice](https:\/\/mathvault.ca\/dmca\/)\n\n- 100shares\n\n[Math Vault](https:\/\/mathvault.ca\/?blackhole=cb4b0d2f5d \"Do NOT follow this link or you will be banned from the site!\")\n\nNotifications","attrs_readable_markdown":"Let us take a moment to ponder how truly bizarre the **[Laplace transform](https:\/\/mathvault.ca\/hub\/higher-math\/math-symbols\/calculus-analysis-symbols\/#Key_Transforms)** is.\n\nYou put in a sine and get an oddly simple, **arbitrary-looking fraction**. Why do we suddenly have squares?\n\nYou look at the table of **common Laplace transforms** to find a pattern and you see no rhyme, no reason, no obvious link between different functions and their different, very different, results.\n\nWhat’s going on here?\n\nOr so we thought when we first encountered the cursive L in school.\n\n![Laplace transform of function f](https:\/\/mathvault.ca\/wp-content\/uploads\/Laplace-Transform.png)\n\nTable of Contents\n\n- [What does the Laplace transform do, really?](https:\/\/mathvault.ca\/laplace-transform\/#What_does_the_Laplace_transform_do_really)\n- [Some Preliminary Examples](https:\/\/mathvault.ca\/laplace-transform\/#Some_Preliminary_Examples)\n- [Looking Inside the Laplace Transform of Sine](https:\/\/mathvault.ca\/laplace-transform\/#Looking_Inside_the_Laplace_Transform_of_Sine)\n- [Diverging Functions: What the Laplace Transform is for](https:\/\/mathvault.ca\/laplace-transform\/#Diverging_Functions_What_the_Laplace_Transform_is_for)\n- [A Transform of Unfathomable Power](https:\/\/mathvault.ca\/laplace-transform\/#A_Transform_of_Unfathomable_Power)\n## [What does the Laplace transform do, really?](https:\/\/mathvault.ca\/laplace-transform\/#toc)\nAt a high level, Laplace transform is an **[integral transform](https:\/\/en.wikipedia.org\/wiki\/Integral_transform#:~:text=In%20mathematics%2C%20an%20integral%20transform,in%20the%20original%20function%20space.)** mostly encountered in differential equations — in electrical engineering for instance — where electric circuits are represented as differential equations.\n\nIn fact, it takes a **time-domain function**, where t is the variable, and outputs a **frequency-domain function**, where s is the variable. Definition-wise, Laplace transform takes a function of real variable f ( t ) (defined for all t ≥ 0) to a function of complex variable F ( s ) as follows:\n\nL\n\n{\n\nf\n\n(\n\nt\n\n)\n\n}\n\n\\=\n\n∫\n\n0\n\n∞\n\nf\n\n(\n\nt\n\n)\n\ne\n\n−\n\ns\n\nt\n\nd\n\nt\n\n\\=\n\nF\n\n(\n\ns\n\n)\n## [Some Preliminary Examples](https:\/\/mathvault.ca\/laplace-transform\/#toc)\nWhat fate awaits **simple functions** as they enter the Laplace transform?\n\nTake the simplest function: the **constant function** f ( t ) \\= 1. In this case, putting 1 in the transform yields 1 \/ s, which means that we went from a constant to a variable-dependent function.\n\n(Odd but not too worrying. After all, we’ve seen 1 \/ x integrating to ln ⁡ x in calculus. Not a constant-to-variable situation of course, but an unexpected transformation nonetheless.)\n\nLet us take it up a notch, with the **linear function** f ( t ) \\= t. After the transformation, it is turned into 1 \/ s 2, which means that we went from 1 → 1 \/ s to t → 1 \/ s 2. A pattern begins to emerge.\n\nNow what about f ( t ) \\= t n? With this simple **power function**, we end up with: L { t n } \\= n \\! s n \\+ 1 So there was a factorial in L { t } all along, hidden by the fact that 1 \\! \\= 1. What else is the transform hiding?\n\nHere, a glance at a table of **[common Laplace transforms](https:\/\/tutorial.math.lamar.edu\/pdf\/Laplace_Table.pdf)** would show that the emerging pattern cannot explain other functions easily. Things get weird, and the weirdness escalates quickly — which brings us back to the sine function.\n\n## [Looking Inside the Laplace Transform of Sine](https:\/\/mathvault.ca\/laplace-transform\/#toc)\nLet us unpack what happens to our sine function as we Laplace-transform it. We begin by noticing that a sine function can be expressed as a **complex exponential** — an indirect result of the celebrated [Euler’s formula](https:\/\/mathvault.ca\/euler-formula\/):e i t \\= cos ⁡ t \\+ i sin ⁡ tIn fact, a sine is often expressed in terms of exponentials for **ease of calculation**, so if we apply that to the function f ( t ) \\= sin ⁡ ( a t ), we would get: sin ⁡ ( a t ) \\= e i a t − e − i a t 2 iThus the **Laplace transform** of sin ⁡ ( a t ) then becomes:  \nL { sin ⁡ ( a t ) } \\= 1 2 i ∫ 0 ∞ ( e i a t − e − i a t ) e − s t d twhich means that we have a **product of exponentials**. Distributing the terms, we get:  \nL { sin ⁡ ( a t ) } \\= 1 2 i ∫ 0 ∞ e i a t − s t − e − i a t − s t d t\n\nHere, **factoring** the t in the exponents yields:  \nL { sin ⁡ ( a t ) } \\= 1 2 i ∫ 0 ∞ e ( i a − s ) t − e ( − i a − s ) t d tand since Re ( s ) \\> 0 by assumption, we can proceed with the integration from 0 to ∞ as usual:  \nL { sin ⁡ ( a t ) } \\= e ( i a − s ) t 2 i ( i a − s ) \\| 0 ∞ − e ( − i a − s ) t 2 i ( − i a − s ) \\| 0 ∞\n\nLet us simplify further. **Distributing** the i inside the parentheses, we get:  \nL { sin ⁡ ( a t ) } \\= e ( i a − s ) t 2 ( − a − i s ) \\| 0 ∞ − e ( − i a − s ) t 2 ( a − i s ) \\| 0 ∞By evaluating the t at the **boundaries**, we get:  \nL { sin ⁡ ( a t ) } \\= ( e ( i a − s ) ⋅ ∞ 2 ( − a − i s ) − e ( i a − s ) ⋅ 0 2 ( − a − i s ) ) − ( e ( − i a − s ) ⋅ ∞ 2 ( a − i s ) − e ( − i a − s ) ⋅ 0 2 ( a − i s ) )And because Re ( s ) \\> 0 by assumption, both e ( i a − s ) ⋅ ∞ and e ( − i a − s ) ⋅ ∞ oscillate to 0 (i.e., **[vanish at infinity](https:\/\/mathvault.ca\/math-glossary\/#vanish)**), after which we are then left with:L { sin ⁡ ( a t ) } \\= 1 − 2 ( − a − i s ) \\+ 1 2 ( a − i s )Once there, merging the **fractions** together would yield:L { sin ⁡ ( a t ) } \\= 2 ( a − i s ) − 2 ( − a − i s ) − 4 ( a − i s ) ( − a − i s ) \\= 2 a − 2 i s \\+ 2 a \\+ 2 i s 4 ( a 2 \\+ i s a − i s a \\+ s 2 ) \\= 4 a 4 ( a 2 \\+ s 2 ) \\= a a 2 \\+ s 2which shows that after Laplace transform, a sine is turned into a more tractable **geometric function**. By following similar reasoning, the Laplace transform of cosine can be shown to be equal to the following expression as well: L { cos ⁡ ( a t ) } \\= s a 2 \\+ s 2 ( Re ( s ) \\> 0 ) But then, one might argue “Why do we need to transform trigonometric functions like this when we can just **integrate** them?”\n\n## [Diverging Functions: What the Laplace Transform is for](https:\/\/mathvault.ca\/laplace-transform\/#toc)\nWhat if we throw a wrench in there by introducing a **diverging function**, say, f ( t ) \\= e a t? As it turns out, the Laplace transform of the exponential e a t is actually deceptively simple: L { e a t } \\= ∫ 0 ∞ e a t e − s t d t \\= ∫ 0 ∞ e ( a − s ) t d tHere, we see that so long as Re ( s ) \\> a, we would get that: ∫ 0 ∞ e ( a − s ) t d t \\= e ( a − s ) t a − s \\| 0 ∞ \\= 0 − 1 a − s \\= 1 s − a That is, as long as Re ( s ) \\> a, the Laplace transform of e a t is a simple 1 \/ ( s − a ). Here’s a **video version** of the derivation for the record.\n\nOn the other hand, if we mix the exponential e a t with the **power function** t n, we would then have: L { t n e a t } \\= ∫ 0 ∞ t n e a t e − s t d t which, after a bit of [recursion](https:\/\/mathvault.ca\/math-glossary\/#recursion) and [integration by parts](https:\/\/mathvault.ca\/integration-overshooting-method\/#Integration_By_Parts), would become:n \\! ( s − a ) n \\+ 1Here, notice how the transforms of exponential and power function are both **represented** in the expression, with the factorial n \\!, the 1 \/ ( s − a ) fraction, and the n \\+ 1 exponent.\n\nIn fact, it turns out that we can integrate *any* function with the Laplace transform, as long as it does not **diverge** faster than the e a t exponential. In the tables of Laplace transforms, you might have noticed the Re ( s ) \\> a condition. That is what the condition is alluding to.\n\n## [A Transform of Unfathomable Power](https:\/\/mathvault.ca\/laplace-transform\/#toc)\nHowever, what we have seen is only the tip of the iceberg, since we can also use Laplace transform to transform the **derivatives** as well. In goes f ( n ) ( t ). Something happens. Then out goes:s n L { f ( t ) } − ∑ r \\= 0 n − 1 s n − 1 − r f ( r ) ( 0 )For example, when n \\= 2, we have that:L { f ′ ′ ( t ) } \\= s 2 L { f ( t ) } − s f ( 0 ) − f ′ ( 0 )In addition to the derivatives, the L can also process some **integrals**: the integral sine, cosine and exponential, as well as the [error function](https:\/\/mathvault.ca\/hub\/higher-math\/math-symbols\/probability-statistics-symbols\/#Continuous_Probability_Distributions_and_Associated_Functions) — to name a few.\n\nBut that’s not all. There is also the **[inverse Laplace transform](https:\/\/mathvault.ca\/hub\/higher-math\/math-symbols\/calculus-analysis-symbols\/#Key_Transforms)**, which takes a frequency-domain function and renders a time-domain function.\n\nIn fact, performing the transform from time to frequency and back once introduces a factor of 1 \/ 2 π. Sometimes, you’ll see the whole fraction in front of the inverse function, while other times, the transform and its inverse share a factor of 1 \/ 2 π.\n\nThis is as if the Kraken could restitute the boat intact — but only for a factor of 1 \/ 2 π.\n\nThe Laplace transform, even after all those years, never ceases to bring us awe with its power. Here’s a **table** summarizing the transforms we’ve discussed thus far:\n\n| Function | Laplace Transform |\n|---|---|\n| 1 | 1 s |\n| t | 1 s 2 |\n| t n | n \\! s n \\+ 1 |\n| e a t | 1 s − a |\n| sin ⁡ ( a t ) | a a 2 \\+ s 2 |\n| cos ⁡ ( a t ) | s a 2 \\+ s 2 |\n| t n e a t | n \\! ( s − a ) n \\+ 1 |\n| f ( 2 ) ( t ) | s 2 L { f ( t ) } − s f ( 0 ) − f ′ ( 0 ) |\n| f ( n ) ( t ) | s n L { f ( t ) } − ∑ r \\= 0 n − 1 s n − 1 − r f ( r ) ( 0 ) |","meta_canonical":null}

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Boilerpipe Text	Let us take a moment to ponder how truly bizarre the Laplace transform is. You put in a sine and get an oddly simple, arbitrary-looking fraction . Why do we suddenly have squares? You look at the table of common Laplace transforms to find a pattern and you see no rhyme, no reason, no obvious link between different functions and their different, very different, results. What’s going on here? Or so we thought when we first encountered the cursive L in school. Table of Contents What does the Laplace transform do, really? Some Preliminary Examples Looking Inside the Laplace Transform of Sine Diverging Functions: What the Laplace Transform is for A Transform of Unfathomable Power What does the Laplace transform do, really? At a high level, Laplace transform is an integral transform mostly encountered in differential equations — in electrical engineering for instance — where electric circuits are represented as differential equations. In fact, it takes a time-domain function , where t is the variable, and outputs a frequency-domain function , where s is the variable. Definition-wise, Laplace transform takes a function of real variable f ( t ) (defined for all t ≥ 0 ) to a function of complex variable F ( s ) as follows: L { f ( t ) } = ∫ 0 ∞ f ( t ) e − s t d t = F ( s ) Some Preliminary Examples What fate awaits simple functions as they enter the Laplace transform? Take the simplest function: the constant function f ( t ) = 1 . In this case, putting 1 in the transform yields 1 / s , which means that we went from a constant to a variable-dependent function. (Odd but not too worrying. After all, we’ve seen 1 / x integrating to ln ⁡ x in calculus. Not a constant-to-variable situation of course, but an unexpected transformation nonetheless.) Let us take it up a notch, with the linear function f ( t ) = t . After the transformation, it is turned into 1 / s 2 , which means that we went from 1 → 1 / s to t → 1 / s 2 . A pattern begins to emerge. Now what about f ( t ) = t n ? With this simple power function , we end up with: L { t n } = n ! s n + 1 So there was a factorial in L { t } all along, hidden by the fact that 1 ! = 1 . What else is the transform hiding? Here, a glance at a table of common Laplace transforms would show that the emerging pattern cannot explain other functions easily. Things get weird, and the weirdness escalates quickly — which brings us back to the sine function. Looking Inside the Laplace Transform of Sine Let us unpack what happens to our sine function as we Laplace-transform it. We begin by noticing that a sine function can be expressed as a complex exponential — an indirect result of the celebrated Euler’s formula : e i t = cos ⁡ t + i sin ⁡ t In fact, a sine is often expressed in terms of exponentials for ease of calculation , so if we apply that to the function f ( t ) = sin ⁡ ( a t ) , we would get: sin ⁡ ( a t ) = e i a t − e − i a t 2 i Thus the Laplace transform of sin ⁡ ( a t ) then becomes: L { sin ⁡ ( a t ) } = 1 2 i ∫ 0 ∞ ( e i a t − e − i a t ) e − s t d t which means that we have a product of exponentials . Distributing the terms, we get: L { sin ⁡ ( a t ) } = 1 2 i ∫ 0 ∞ e i a t − s t − e − i a t − s t d t Here, factoring the t in the exponents yields: L { sin ⁡ ( a t ) } = 1 2 i ∫ 0 ∞ e ( i a − s ) t − e ( − i a − s ) t d t and since Re ( s ) > 0 by assumption, we can proceed with the integration from 0 to ∞ as usual: L { sin ⁡ ( a t ) } = e ( i a − s ) t 2 i ( i a − s ) \| 0 ∞ − e ( − i a − s ) t 2 i ( − i a − s ) \| 0 ∞ Let us simplify further. Distributing the i inside the parentheses, we get: L { sin ⁡ ( a t ) } = e ( i a − s ) t 2 ( − a − i s ) \| 0 ∞ − e ( − i a − s ) t 2 ( a − i s ) \| 0 ∞ By evaluating the t at the boundaries , we get: L { sin ⁡ ( a t ) } = ( e ( i a − s ) ⋅ ∞ 2 ( − a − i s ) − e ( i a − s ) ⋅ 0 2 ( − a − i s ) ) − ( e ( − i a − s ) ⋅ ∞ 2 ( a − i s ) − e ( − i a − s ) ⋅ 0 2 ( a − i s ) ) And because Re ( s ) > 0 by assumption, both e ( i a − s ) ⋅ ∞ and e ( − i a − s ) ⋅ ∞ oscillate to 0 (i.e., vanish at infinity ), after which we are then left with: L { sin ⁡ ( a t ) } = 1 − 2 ( − a − i s ) + 1 2 ( a − i s ) Once there, merging the fractions together would yield: L { sin ⁡ ( a t ) } = 2 ( a − i s ) − 2 ( − a − i s ) − 4 ( a − i s ) ( − a − i s ) = 2 a − 2 i s + 2 a + 2 i s 4 ( a 2 + i s a − i s a + s 2 ) = 4 a 4 ( a 2 + s 2 ) = a a 2 + s 2 which shows that after Laplace transform, a sine is turned into a more tractable geometric function . By following similar reasoning, the Laplace transform of cosine can be shown to be equal to the following expression as well: L { cos ⁡ ( a t ) } = s a 2 + s 2 ( Re ( s ) > 0 ) But then, one might argue “Why do we need to transform trigonometric functions like this when we can just integrate them?” Diverging Functions: What the Laplace Transform is for What if we throw a wrench in there by introducing a diverging function , say, f ( t ) = e a t ? As it turns out, the Laplace transform of the exponential e a t is actually deceptively simple: L { e a t } = ∫ 0 ∞ e a t e − s t d t = ∫ 0 ∞ e ( a − s ) t d t Here, we see that so long as Re ( s ) > a , we would get that: ∫ 0 ∞ e ( a − s ) t d t = e ( a − s ) t a − s \| 0 ∞ = 0 − 1 a − s = 1 s − a That is, as long as Re ( s ) > a , the Laplace transform of e a t is a simple 1 / ( s − a ) . Here’s a video version of the derivation for the record. On the other hand, if we mix the exponential e a t with the power function t n , we would then have: L { t n e a t } = ∫ 0 ∞ t n e a t e − s t d t which, after a bit of recursion and integration by parts , would become: n ! ( s − a ) n + 1 Here, notice how the transforms of exponential and power function are both represented in the expression, with the factorial n ! , the 1 / ( s − a ) fraction, and the n + 1 exponent. In fact, it turns out that we can integrate any function with the Laplace transform, as long as it does not diverge faster than the e a t exponential. In the tables of Laplace transforms, you might have noticed the Re ( s ) > a condition. That is what the condition is alluding to. A Transform of Unfathomable Power However, what we have seen is only the tip of the iceberg, since we can also use Laplace transform to transform the derivatives as well. In goes f ( n ) ( t ) . Something happens. Then out goes: s n L { f ( t ) } − ∑ r = 0 n − 1 s n − 1 − r f ( r ) ( 0 ) For example, when n = 2 , we have that: L { f ′ ′ ( t ) } = s 2 L { f ( t ) } − s f ( 0 ) − f ′ ( 0 ) In addition to the derivatives, the L can also process some integrals : the integral sine, cosine and exponential, as well as the error function — to name a few. But that’s not all. There is also the inverse Laplace transform , which takes a frequency-domain function and renders a time-domain function. In fact, performing the transform from time to frequency and back once introduces a factor of 1 / 2 π . Sometimes, you’ll see the whole fraction in front of the inverse function, while other times, the transform and its inverse share a factor of 1 / 2 π . This is as if the Kraken could restitute the boat intact — but only for a factor of 1 / 2 π . The Laplace transform, even after all those years, never ceases to bring us awe with its power. Here’s a table summarizing the transforms we’ve discussed thus far: Function Laplace Transform 1 1 s t 1 s 2 t n n ! s n + 1 e a t 1 s − a sin ⁡ ( a t ) a a 2 + s 2 cos ⁡ ( a t ) s a 2 + s 2 t n e a t n ! ( s − a ) n + 1 f ( 2 ) ( t ) s 2 L { f ( t ) } − s f ( 0 ) − f ′ ( 0 ) f ( n ) ( t ) s n L { f ( t ) } − ∑ r = 0 n − 1 s n − 1 − r f ( r ) ( 0 )
Markdown	![](https://www.facebook.com/tr?id=3257118080998043&ev=PageView&noscript=1) [![Light logo of Math Vault, an online resource hub of higher mathematics](https://mathvault.ca/wp-content/uploads/New-MVLogo-White.png)](https://mathvault.ca/) - [Learn](https://mathvault.ca/laplace-transform/) - [Hub](https://mathvault.ca/hub/) Definitive resource hub on everything higher math - [Vault](https://mathvault.ca/vault/) Bonus guides and lessons on mathematics and other related topics - [Products & Services](https://mathvault.ca/shop) - [Definitive Guide to Learning Higher Mathematics](https://mathvault.ca/higher-math-learning-guide/) - [Ultimate LaTeX Reference Guide](https://mathvault.ca/latex/) - [Comprehensive List of Mathematical Symbols](https://mathvault.ca/product/comprehensive-list-of-mathematical-symbols/) - [Math Tutoring / Consulting](https://mathvault.ca/tutoring/) - [Resources]() - [Higher Math Proficiency Test](https://mathvault.ca/math-test/) - [10 Commandments of Higher Math Learning](https://mathvault.ca/10-commandments/) - [Math Vault Linear Algebra Ebook Series](https://mathvault.ca/free-ebooks-series/) - [Recommended Math Books](https://mathvault.ca/books/) - [Recommended Math Websites](https://mathvault.ca/websites/) - [Recommended Online Tools](https://mathvault.ca/online-tools/) - [About]() - [About Us](https://mathvault.ca/about/) Where we came from, and where we're going - [Write for Us](https://mathvault.ca/write-for-us/) Join us in contributing to the glory of mathematics [Start Here](https://mathvault.ca/introduction-series/) [Kim Thibault](https://mathvault.ca/author/kimthibo/ "Kim Thibault") # Laplace Transform: A First Introduction College Math, Calculus, Complex Number - [Home](https://mathvault.ca/) - » - [Vault](https://mathvault.ca/vault/) - » - [College Math](https://mathvault.ca/college-math/) - » - Laplace Transform: A First Introduction Let us take a moment to ponder how truly bizarre the [Laplace transform](https://mathvault.ca/hub/higher-math/math-symbols/calculus-analysis-symbols/#Key_Transforms) is. You put in a sine and get an oddly simple, arbitrary-looking fraction. Why do we suddenly have squares? You look at the table of common Laplace transforms to find a pattern and you see no rhyme, no reason, no obvious link between different functions and their different, very different, results. What’s going on here? Or so we thought when we first encountered the cursive L in school. ![Laplace transform of function f](https://mathvault.ca/wp-content/uploads/Laplace-Transform.png) Table of Contents [Toggle](https://mathvault.ca/laplace-transform/) - [What does the Laplace transform do, really?](https://mathvault.ca/laplace-transform/#What_does_the_Laplace_transform_do_really) - [Some Preliminary Examples](https://mathvault.ca/laplace-transform/#Some_Preliminary_Examples) - [Looking Inside the Laplace Transform of Sine](https://mathvault.ca/laplace-transform/#Looking_Inside_the_Laplace_Transform_of_Sine) - [Diverging Functions: What the Laplace Transform is for](https://mathvault.ca/laplace-transform/#Diverging_Functions_What_the_Laplace_Transform_is_for) - [A Transform of Unfathomable Power](https://mathvault.ca/laplace-transform/#A_Transform_of_Unfathomable_Power) ## [What does the Laplace transform do, really?](https://mathvault.ca/laplace-transform/#toc) At a high level, Laplace transform is an [integral transform](https://en.wikipedia.org/wiki/Integral_transform#:~:text=In%20mathematics%2C%20an%20integral%20transform,in%20the%20original%20function%20space.) mostly encountered in differential equations — in electrical engineering for instance — where electric circuits are represented as differential equations. In fact, it takes a time-domain function, where t is the variable, and outputs a frequency-domain function, where s is the variable. Definition-wise, Laplace transform takes a function of real variable f ( t ) (defined for all t ≥ 0) to a function of complex variable F ( s ) as follows: L { f ( t ) } \= ∫ 0 ∞ f ( t ) e − s t d t \= F ( s ) ## [Some Preliminary Examples](https://mathvault.ca/laplace-transform/#toc) What fate awaits simple functions as they enter the Laplace transform? Take the simplest function: the constant function f ( t ) \= 1. In this case, putting 1 in the transform yields 1 / s, which means that we went from a constant to a variable-dependent function. (Odd but not too worrying. After all, we’ve seen 1 / x integrating to ln ⁡ x in calculus. Not a constant-to-variable situation of course, but an unexpected transformation nonetheless.) Let us take it up a notch, with the linear function f ( t ) \= t. After the transformation, it is turned into 1 / s 2, which means that we went from 1 → 1 / s to t → 1 / s 2. A pattern begins to emerge. Now what about f ( t ) \= t n? With this simple power function, we end up with: L { t n } \= n \! s n \+ 1 So there was a factorial in L { t } all along, hidden by the fact that 1 \! \= 1. What else is the transform hiding? Here, a glance at a table of [common Laplace transforms](https://tutorial.math.lamar.edu/pdf/Laplace_Table.pdf) would show that the emerging pattern cannot explain other functions easily. Things get weird, and the weirdness escalates quickly — which brings us back to the sine function. ## [Looking Inside the Laplace Transform of Sine](https://mathvault.ca/laplace-transform/#toc) Let us unpack what happens to our sine function as we Laplace-transform it. We begin by noticing that a sine function can be expressed as a complex exponential — an indirect result of the celebrated [Euler’s formula](https://mathvault.ca/euler-formula/):e i t \= cos ⁡ t \+ i sin ⁡ tIn fact, a sine is often expressed in terms of exponentials for ease of calculation, so if we apply that to the function f ( t ) \= sin ⁡ ( a t ), we would get: sin ⁡ ( a t ) \= e i a t − e − i a t 2 iThus the Laplace transform of sin ⁡ ( a t ) then becomes: L { sin ⁡ ( a t ) } \= 1 2 i ∫ 0 ∞ ( e i a t − e − i a t ) e − s t d twhich means that we have a product of exponentials. Distributing the terms, we get: L { sin ⁡ ( a t ) } \= 1 2 i ∫ 0 ∞ e i a t − s t − e − i a t − s t d t Here, factoring the t in the exponents yields: L { sin ⁡ ( a t ) } \= 1 2 i ∫ 0 ∞ e ( i a − s ) t − e ( − i a − s ) t d tand since Re ( s ) \> 0 by assumption, we can proceed with the integration from 0 to ∞ as usual: L { sin ⁡ ( a t ) } \= e ( i a − s ) t 2 i ( i a − s ) \\| 0 ∞ − e ( − i a − s ) t 2 i ( − i a − s ) \\| 0 ∞ Let us simplify further. Distributing the i inside the parentheses, we get: L { sin ⁡ ( a t ) } \= e ( i a − s ) t 2 ( − a − i s ) \\| 0 ∞ − e ( − i a − s ) t 2 ( a − i s ) \\| 0 ∞By evaluating the t at the boundaries, we get: L { sin ⁡ ( a t ) } \= ( e ( i a − s ) ⋅ ∞ 2 ( − a − i s ) − e ( i a − s ) ⋅ 0 2 ( − a − i s ) ) − ( e ( − i a − s ) ⋅ ∞ 2 ( a − i s ) − e ( − i a − s ) ⋅ 0 2 ( a − i s ) )And because Re ( s ) \> 0 by assumption, both e ( i a − s ) ⋅ ∞ and e ( − i a − s ) ⋅ ∞ oscillate to 0 (i.e., [vanish at infinity](https://mathvault.ca/math-glossary/#vanish)), after which we are then left with:L { sin ⁡ ( a t ) } \= 1 − 2 ( − a − i s ) \+ 1 2 ( a − i s )Once there, merging the fractions together would yield:L { sin ⁡ ( a t ) } \= 2 ( a − i s ) − 2 ( − a − i s ) − 4 ( a − i s ) ( − a − i s ) \= 2 a − 2 i s \+ 2 a \+ 2 i s 4 ( a 2 \+ i s a − i s a \+ s 2 ) \= 4 a 4 ( a 2 \+ s 2 ) \= a a 2 \+ s 2which shows that after Laplace transform, a sine is turned into a more tractable geometric function. By following similar reasoning, the Laplace transform of cosine can be shown to be equal to the following expression as well: L { cos ⁡ ( a t ) } \= s a 2 \+ s 2 ( Re ( s ) \> 0 ) But then, one might argue “Why do we need to transform trigonometric functions like this when we can just integrate them?” ## [Diverging Functions: What the Laplace Transform is for](https://mathvault.ca/laplace-transform/#toc) What if we throw a wrench in there by introducing a diverging function, say, f ( t ) \= e a t? As it turns out, the Laplace transform of the exponential e a t is actually deceptively simple: L { e a t } \= ∫ 0 ∞ e a t e − s t d t \= ∫ 0 ∞ e ( a − s ) t d tHere, we see that so long as Re ( s ) \> a, we would get that: ∫ 0 ∞ e ( a − s ) t d t \= e ( a − s ) t a − s \\| 0 ∞ \= 0 − 1 a − s \= 1 s − a That is, as long as Re ( s ) \> a, the Laplace transform of e a t is a simple 1 / ( s − a ). Here’s a video version of the derivation for the record. On the other hand, if we mix the exponential e a t with the power function t n, we would then have: L { t n e a t } \= ∫ 0 ∞ t n e a t e − s t d t which, after a bit of [recursion](https://mathvault.ca/math-glossary/#recursion) and [integration by parts](https://mathvault.ca/integration-overshooting-method/#Integration_By_Parts), would become:n \! ( s − a ) n \+ 1Here, notice how the transforms of exponential and power function are both represented in the expression, with the factorial n \!, the 1 / ( s − a ) fraction, and the n \+ 1 exponent. In fact, it turns out that we can integrate any function with the Laplace transform, as long as it does not diverge faster than the e a t exponential. In the tables of Laplace transforms, you might have noticed the Re ( s ) \> a condition. That is what the condition is alluding to. ## [A Transform of Unfathomable Power](https://mathvault.ca/laplace-transform/#toc) However, what we have seen is only the tip of the iceberg, since we can also use Laplace transform to transform the derivatives as well. In goes f ( n ) ( t ). Something happens. Then out goes:s n L { f ( t ) } − ∑ r \= 0 n − 1 s n − 1 − r f ( r ) ( 0 )For example, when n \= 2, we have that:L { f ′ ′ ( t ) } \= s 2 L { f ( t ) } − s f ( 0 ) − f ′ ( 0 )In addition to the derivatives, the L can also process some integrals: the integral sine, cosine and exponential, as well as the [error function](https://mathvault.ca/hub/higher-math/math-symbols/probability-statistics-symbols/#Continuous_Probability_Distributions_and_Associated_Functions) — to name a few. But that’s not all. There is also the [inverse Laplace transform](https://mathvault.ca/hub/higher-math/math-symbols/calculus-analysis-symbols/#Key_Transforms), which takes a frequency-domain function and renders a time-domain function. In fact, performing the transform from time to frequency and back once introduces a factor of 1 / 2 π. Sometimes, you’ll see the whole fraction in front of the inverse function, while other times, the transform and its inverse share a factor of 1 / 2 π. This is as if the Kraken could restitute the boat intact — but only for a factor of 1 / 2 π. The Laplace transform, even after all those years, never ceases to bring us awe with its power. Here’s a table summarizing the transforms we’ve discussed thus far: \| Function \| Laplace Transform \| \|---\|---\| \| 1 \| 1 s \| \| t \| 1 s 2 \| \| t n \| n \! s n \+ 1 \| \| e a t \| 1 s − a \| \| sin ⁡ ( a t ) \| a a 2 \+ s 2 \| \| cos ⁡ ( a t ) \| s a 2 \+ s 2 \| \| t n e a t \| n \! ( s − a ) n \+ 1 \| \| f ( 2 ) ( t ) \| s 2 L { f ( t ) } − s f ( 0 ) − f ′ ( 0 ) \| \| f ( n ) ( t ) \| s n L { f ( t ) } − ∑ r \= 0 n − 1 s n − 1 − r f ( r ) ( 0 ) \| [![The Definitive Guide to Higher Mathematics — Cover](https://mathvault.ca/wp-content/uploads/Higher-Math-Guide-Cover.png)](https://mathvault.ca/higher-math-learning-guide/) How's your higher math going? Shallow learning and mechanical practices rarely work in higher mathematics. Instead, use these 10 principles to optimize your learning and prevent years of wasted effort. [Hmm... Tell me more.](https://mathvault.ca/higher-math-learning-guide/) - 100shares ![Math Vault Standard Post](https://mathvault.ca/wp-content/litespeed/avatar/8f28b1715d6e47ed7ec221f47fa20fc9.jpg?ver=1775679107) Kim Thibault #### About the author Kim Thibault is an incorrigible polymath. After a Ph.D. in Physics, she did applied research in machine learning for audio, then a stint in programming, to finally become an author and scientific translator. She occasionally solves differential equations as a hobby. Find her on her [website](http://kimthibault.com/) and on [Linkedin](https://linkedin.com/in/kimthibaultphd). 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[Reply](https://mathvault.ca/laplace-transform/#comment-3586) 1. [![](https://mathvault.ca/wp-content/litespeed/avatar/d96413892cebda856b0216ca7e7f9f76.jpg?ver=1775604434)](https://mathvault.ca/) [Math Vault](https://mathvault.ca/) says: [February 10, 2021 at 2:02 PM](https://mathvault.ca/laplace-transform/#comment-3587) Hi Amadi. Thanks for your interest. If you click on the “Print” button in your browser, you can save the guide as a PDF. That would be a way of generating a soft copy. 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Readable Markdown	Let us take a moment to ponder how truly bizarre the [Laplace transform](https://mathvault.ca/hub/higher-math/math-symbols/calculus-analysis-symbols/#Key_Transforms) is. You put in a sine and get an oddly simple, arbitrary-looking fraction. Why do we suddenly have squares? You look at the table of common Laplace transforms to find a pattern and you see no rhyme, no reason, no obvious link between different functions and their different, very different, results. What’s going on here? Or so we thought when we first encountered the cursive L in school. ![Laplace transform of function f](https://mathvault.ca/wp-content/uploads/Laplace-Transform.png) Table of Contents - [What does the Laplace transform do, really?](https://mathvault.ca/laplace-transform/#What_does_the_Laplace_transform_do_really) - [Some Preliminary Examples](https://mathvault.ca/laplace-transform/#Some_Preliminary_Examples) - [Looking Inside the Laplace Transform of Sine](https://mathvault.ca/laplace-transform/#Looking_Inside_the_Laplace_Transform_of_Sine) - [Diverging Functions: What the Laplace Transform is for](https://mathvault.ca/laplace-transform/#Diverging_Functions_What_the_Laplace_Transform_is_for) - [A Transform of Unfathomable Power](https://mathvault.ca/laplace-transform/#A_Transform_of_Unfathomable_Power) ## [What does the Laplace transform do, really?](https://mathvault.ca/laplace-transform/#toc) At a high level, Laplace transform is an [integral transform](https://en.wikipedia.org/wiki/Integral_transform#:~:text=In%20mathematics%2C%20an%20integral%20transform,in%20the%20original%20function%20space.) mostly encountered in differential equations — in electrical engineering for instance — where electric circuits are represented as differential equations. In fact, it takes a time-domain function, where t is the variable, and outputs a frequency-domain function, where s is the variable. Definition-wise, Laplace transform takes a function of real variable f ( t ) (defined for all t ≥ 0) to a function of complex variable F ( s ) as follows: L { f ( t ) } \= ∫ 0 ∞ f ( t ) e − s t d t \= F ( s ) ## [Some Preliminary Examples](https://mathvault.ca/laplace-transform/#toc) What fate awaits simple functions as they enter the Laplace transform? Take the simplest function: the constant function f ( t ) \= 1. In this case, putting 1 in the transform yields 1 / s, which means that we went from a constant to a variable-dependent function. (Odd but not too worrying. After all, we’ve seen 1 / x integrating to ln ⁡ x in calculus. Not a constant-to-variable situation of course, but an unexpected transformation nonetheless.) Let us take it up a notch, with the linear function f ( t ) \= t. After the transformation, it is turned into 1 / s 2, which means that we went from 1 → 1 / s to t → 1 / s 2. A pattern begins to emerge. Now what about f ( t ) \= t n? With this simple power function, we end up with: L { t n } \= n \! s n \+ 1 So there was a factorial in L { t } all along, hidden by the fact that 1 \! \= 1. What else is the transform hiding? Here, a glance at a table of [common Laplace transforms](https://tutorial.math.lamar.edu/pdf/Laplace_Table.pdf) would show that the emerging pattern cannot explain other functions easily. Things get weird, and the weirdness escalates quickly — which brings us back to the sine function. ## [Looking Inside the Laplace Transform of Sine](https://mathvault.ca/laplace-transform/#toc) Let us unpack what happens to our sine function as we Laplace-transform it. We begin by noticing that a sine function can be expressed as a complex exponential — an indirect result of the celebrated [Euler’s formula](https://mathvault.ca/euler-formula/):e i t \= cos ⁡ t \+ i sin ⁡ tIn fact, a sine is often expressed in terms of exponentials for ease of calculation, so if we apply that to the function f ( t ) \= sin ⁡ ( a t ), we would get: sin ⁡ ( a t ) \= e i a t − e − i a t 2 iThus the Laplace transform of sin ⁡ ( a t ) then becomes: L { sin ⁡ ( a t ) } \= 1 2 i ∫ 0 ∞ ( e i a t − e − i a t ) e − s t d twhich means that we have a product of exponentials. Distributing the terms, we get: L { sin ⁡ ( a t ) } \= 1 2 i ∫ 0 ∞ e i a t − s t − e − i a t − s t d t Here, factoring the t in the exponents yields: L { sin ⁡ ( a t ) } \= 1 2 i ∫ 0 ∞ e ( i a − s ) t − e ( − i a − s ) t d tand since Re ( s ) \> 0 by assumption, we can proceed with the integration from 0 to ∞ as usual: L { sin ⁡ ( a t ) } \= e ( i a − s ) t 2 i ( i a − s ) \\| 0 ∞ − e ( − i a − s ) t 2 i ( − i a − s ) \\| 0 ∞ Let us simplify further. Distributing the i inside the parentheses, we get: L { sin ⁡ ( a t ) } \= e ( i a − s ) t 2 ( − a − i s ) \\| 0 ∞ − e ( − i a − s ) t 2 ( a − i s ) \\| 0 ∞By evaluating the t at the boundaries, we get: L { sin ⁡ ( a t ) } \= ( e ( i a − s ) ⋅ ∞ 2 ( − a − i s ) − e ( i a − s ) ⋅ 0 2 ( − a − i s ) ) − ( e ( − i a − s ) ⋅ ∞ 2 ( a − i s ) − e ( − i a − s ) ⋅ 0 2 ( a − i s ) )And because Re ( s ) \> 0 by assumption, both e ( i a − s ) ⋅ ∞ and e ( − i a − s ) ⋅ ∞ oscillate to 0 (i.e., [vanish at infinity](https://mathvault.ca/math-glossary/#vanish)), after which we are then left with:L { sin ⁡ ( a t ) } \= 1 − 2 ( − a − i s ) \+ 1 2 ( a − i s )Once there, merging the fractions together would yield:L { sin ⁡ ( a t ) } \= 2 ( a − i s ) − 2 ( − a − i s ) − 4 ( a − i s ) ( − a − i s ) \= 2 a − 2 i s \+ 2 a \+ 2 i s 4 ( a 2 \+ i s a − i s a \+ s 2 ) \= 4 a 4 ( a 2 \+ s 2 ) \= a a 2 \+ s 2which shows that after Laplace transform, a sine is turned into a more tractable geometric function. By following similar reasoning, the Laplace transform of cosine can be shown to be equal to the following expression as well: L { cos ⁡ ( a t ) } \= s a 2 \+ s 2 ( Re ( s ) \> 0 ) But then, one might argue “Why do we need to transform trigonometric functions like this when we can just integrate them?” ## [Diverging Functions: What the Laplace Transform is for](https://mathvault.ca/laplace-transform/#toc) What if we throw a wrench in there by introducing a diverging function, say, f ( t ) \= e a t? As it turns out, the Laplace transform of the exponential e a t is actually deceptively simple: L { e a t } \= ∫ 0 ∞ e a t e − s t d t \= ∫ 0 ∞ e ( a − s ) t d tHere, we see that so long as Re ( s ) \> a, we would get that: ∫ 0 ∞ e ( a − s ) t d t \= e ( a − s ) t a − s \\| 0 ∞ \= 0 − 1 a − s \= 1 s − a That is, as long as Re ( s ) \> a, the Laplace transform of e a t is a simple 1 / ( s − a ). Here’s a video version of the derivation for the record. On the other hand, if we mix the exponential e a t with the power function t n, we would then have: L { t n e a t } \= ∫ 0 ∞ t n e a t e − s t d t which, after a bit of [recursion](https://mathvault.ca/math-glossary/#recursion) and [integration by parts](https://mathvault.ca/integration-overshooting-method/#Integration_By_Parts), would become:n \! ( s − a ) n \+ 1Here, notice how the transforms of exponential and power function are both represented in the expression, with the factorial n \!, the 1 / ( s − a ) fraction, and the n \+ 1 exponent. In fact, it turns out that we can integrate any function with the Laplace transform, as long as it does not diverge faster than the e a t exponential. In the tables of Laplace transforms, you might have noticed the Re ( s ) \> a condition. That is what the condition is alluding to. ## [A Transform of Unfathomable Power](https://mathvault.ca/laplace-transform/#toc) However, what we have seen is only the tip of the iceberg, since we can also use Laplace transform to transform the derivatives as well. In goes f ( n ) ( t ). Something happens. Then out goes:s n L { f ( t ) } − ∑ r \= 0 n − 1 s n − 1 − r f ( r ) ( 0 )For example, when n \= 2, we have that:L { f ′ ′ ( t ) } \= s 2 L { f ( t ) } − s f ( 0 ) − f ′ ( 0 )In addition to the derivatives, the L can also process some integrals: the integral sine, cosine and exponential, as well as the [error function](https://mathvault.ca/hub/higher-math/math-symbols/probability-statistics-symbols/#Continuous_Probability_Distributions_and_Associated_Functions) — to name a few. But that’s not all. There is also the [inverse Laplace transform](https://mathvault.ca/hub/higher-math/math-symbols/calculus-analysis-symbols/#Key_Transforms), which takes a frequency-domain function and renders a time-domain function. In fact, performing the transform from time to frequency and back once introduces a factor of 1 / 2 π. Sometimes, you’ll see the whole fraction in front of the inverse function, while other times, the transform and its inverse share a factor of 1 / 2 π. This is as if the Kraken could restitute the boat intact — but only for a factor of 1 / 2 π. The Laplace transform, even after all those years, never ceases to bring us awe with its power. Here’s a table summarizing the transforms we’ve discussed thus far: \| Function \| Laplace Transform \| \|---\|---\| \| 1 \| 1 s \| \| t \| 1 s 2 \| \| t n \| n \! s n \+ 1 \| \| e a t \| 1 s − a \| \| sin ⁡ ( a t ) \| a a 2 \+ s 2 \| \| cos ⁡ ( a t ) \| s a 2 \+ s 2 \| \| t n e a t \| n \! ( s − a ) n \+ 1 \| \| f ( 2 ) ( t ) \| s 2 L { f ( t ) } − s f ( 0 ) − f ′ ( 0 ) \| \| f ( n ) ( t ) \| s n L { f ( t ) } − ∑ r \= 0 n − 1 s n − 1 − r f ( r ) ( 0 ) \|
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