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[Jump to content](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#bodyContent) Main menu Main menu move to sidebar hide Navigation - [Main page](https://en.wikipedia.org/wiki/Main_Page "Visit the main page [z]") - [Contents](https://en.wikipedia.org/wiki/Wikipedia:Contents "Guides to browsing Wikipedia") - [Current events](https://en.wikipedia.org/wiki/Portal:Current_events "Articles related to current events") - [Random article](https://en.wikipedia.org/wiki/Special:Random "Visit a randomly selected article [x]") - [About Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:About "Learn about Wikipedia and how it works") - [Contact us](https://en.wikipedia.org/wiki/Wikipedia:Contact_us "How to contact Wikipedia") Contribute - [Help](https://en.wikipedia.org/wiki/Help:Contents "Guidance on how to use and edit Wikipedia") - [Learn to edit](https://en.wikipedia.org/wiki/Help:Introduction "Learn how to edit Wikipedia") - [Community portal](https://en.wikipedia.org/wiki/Wikipedia:Community_portal "The hub for editors") - [Recent changes](https://en.wikipedia.org/wiki/Special:RecentChanges "A list of recent changes to Wikipedia [r]") - [Upload file](https://en.wikipedia.org/wiki/Wikipedia:File_upload_wizard "Add images or other media for use on Wikipedia") - [Special pages](https://en.wikipedia.org/wiki/Special:SpecialPages) [  ](https://en.wikipedia.org/wiki/Main_Page) [Search](https://en.wikipedia.org/wiki/Special:Search "Search Wikipedia [f]") Appearance - [Donate](https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en) - [Create account](https://en.wikipedia.org/w/index.php?title=Special:CreateAccount&returnto=List+of+Laplace+transforms "You are encouraged to create an account and log in; however, it is not mandatory") - [Log in](https://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=List+of+Laplace+transforms "You're encouraged to log in; however, it's not mandatory. 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[4 See also](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#See_also) - [5 References](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#References) Toggle the table of contents # List of Laplace transforms 1 language - [Português](https://pt.wikipedia.org/wiki/Lista_de_transformadas_de_Laplace "Lista de transformadas de Laplace – Portuguese") [Edit links](https://www.wikidata.org/wiki/Special:EntityPage/Q25047839#sitelinks-wikipedia "Edit interlanguage links") - [Article](https://en.wikipedia.org/wiki/List_of_Laplace_transforms "View the content page [c]") - [Talk](https://en.wikipedia.org/wiki/Talk:List_of_Laplace_transforms "Discuss improvements to the content page [t]") English - [Read](https://en.wikipedia.org/wiki/List_of_Laplace_transforms) - [Edit](https://en.wikipedia.org/w/index.php?title=List_of_Laplace_transforms&action=edit "Edit this page [e]") - [View history](https://en.wikipedia.org/w/index.php?title=List_of_Laplace_transforms&action=history "Past revisions of this page [h]") Tools Tools move to sidebar hide Actions - 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[Cite this page](https://en.wikipedia.org/w/index.php?title=Special:CiteThisPage&page=List_of_Laplace_transforms&id=1287808260&wpFormIdentifier=titleform "Information on how to cite this page") - [Get shortened URL](https://en.wikipedia.org/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FList_of_Laplace_transforms) - [Download QR code](https://en.wikipedia.org/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FList_of_Laplace_transforms) Print/export - [Download as PDF](https://en.wikipedia.org/w/index.php?title=Special:DownloadAsPdf&page=List_of_Laplace_transforms&action=show-download-screen "Download this page as a PDF file") - [Printable version](https://en.wikipedia.org/w/index.php?title=List_of_Laplace_transforms&printable=yes "Printable version of this page [p]") In other projects - [Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q25047839 "Structured data on this page hosted by Wikidata [g]") Appearance move to sidebar hide From Wikipedia, the free encyclopedia The following is a **list of Laplace transforms** for many common functions of a single variable.[\[1\]](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_note-1) The [Laplace transform](https://en.wikipedia.org/wiki/Laplace_transform "Laplace transform") is an [integral transform](https://en.wikipedia.org/wiki/Integral_transform "Integral transform") that takes a function of a positive real variable *t* (often time) to a function of a complex variable s (complex [angular frequency](https://en.wikipedia.org/wiki/Angular_frequency "Angular frequency")). ## Properties \[[edit](https://en.wikipedia.org/w/index.php?title=List_of_Laplace_transforms&action=edit&section=1 "Edit section: Properties")\] Main article: [Laplace transform § Properties and theorems](https://en.wikipedia.org/wiki/Laplace_transform#Properties_and_theorems "Laplace transform") The Laplace transform of a function f ( t ) {\\displaystyle f(t)}  can be obtained using the [formal definition](https://en.wikipedia.org/wiki/Laplace_transform#Formal_definition "Laplace transform") of the Laplace transform. However, some properties of the Laplace transform can be used to obtain the Laplace transform of some functions more easily. ### Linearity \[[edit](https://en.wikipedia.org/w/index.php?title=List_of_Laplace_transforms&action=edit&section=2 "Edit section: Linearity")\] For functions f {\\displaystyle f}  and g {\\displaystyle g}  and for scalar a {\\displaystyle a} , the Laplace transform satisfies L { a f ( t ) \+ b g ( t ) } \= a L { f ( t ) } \+ b L { g ( t ) } {\\displaystyle {\\mathcal {L}}\\{af(t)+bg(t)\\}=a{\\mathcal {L}}\\{f(t)\\}+b{\\mathcal {L}}\\{g(t)\\}}  and is, therefore, regarded as a linear operator. ### Time shifting \[[edit](https://en.wikipedia.org/w/index.php?title=List_of_Laplace_transforms&action=edit&section=3 "Edit section: Time shifting")\] The Laplace transform of f ( t − a ) u ( t − a ) {\\displaystyle f(t-a)u(t-a)}  is e − a s F ( s ) {\\displaystyle e^{-as}F(s)} . ### Frequency shifting \[[edit](https://en.wikipedia.org/w/index.php?title=List_of_Laplace_transforms&action=edit&section=4 "Edit section: Frequency shifting")\] The Laplace transform of e a t f ( t ) {\\displaystyle e^{at}f(t)}  is F ( s − a ) {\\displaystyle F(s-a)} . ## Explanatory notes \[[edit](https://en.wikipedia.org/w/index.php?title=List_of_Laplace_transforms&action=edit&section=5 "Edit section: Explanatory notes")\] The unilateral Laplace transform takes as input a function whose time domain is the [non-negative](https://en.wikipedia.org/wiki/Non-negative "Non-negative") reals, which is why all of the time domain functions in the table below are multiples of the [Heaviside step function](https://en.wikipedia.org/wiki/Heaviside_step_function "Heaviside step function"), *u*(*t*). The entries of the table that involve a time delay *τ* are required to be [causal](https://en.wikipedia.org/wiki/Causal_system "Causal system") (meaning that *τ* \> 0). A causal system is a system where the [impulse response](https://en.wikipedia.org/wiki/Impulse_response "Impulse response") *h*(*t*) is zero for all time t prior to *t* = 0. In general, the region of convergence for causal systems is not the same as that of [anticausal systems](https://en.wikipedia.org/wiki/Anticausal_system "Anticausal system"). The following functions and variables are used in the table below: - *δ* represents the [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function "Dirac delta function"). - *u*(*t*) represents the [Heaviside step function](https://en.wikipedia.org/wiki/Heaviside_step_function "Heaviside step function"). Literature may refer to this by other notation, including 1 ( t ) {\\displaystyle 1(t)}  or H ( t ) {\\displaystyle H(t)}  . - Γ(*z*) represents the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function "Gamma function"). - *γ* is the [Euler–Mascheroni constant](https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant "Euler–Mascheroni constant"). - *t* is a [real number](https://en.wikipedia.org/wiki/Real_number "Real number"). It typically represents *time*, although it can represent *any* independent dimension. - *s* is the [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") frequency domain parameter, and Re(*s*) is its [real part](https://en.wikipedia.org/wiki/Real_part "Real part"). - *n* is an [integer](https://en.wikipedia.org/wiki/Integer "Integer"). - *α*, *τ,* and *ω* are real numbers. - *q* is a complex number. ## Table \[[edit](https://en.wikipedia.org/w/index.php?title=List_of_Laplace_transforms&action=edit&section=6 "Edit section: Table")\] | Function | Time domain f ( t ) \= L − 1 { F ( s ) } {\\displaystyle f(t)={\\mathcal {L}}^{-1}\\{F(s)\\}}  | |---|---| ## See also \[[edit](https://en.wikipedia.org/w/index.php?title=List_of_Laplace_transforms&action=edit&section=7 "Edit section: See also")\] - [List of Fourier transforms](https://en.wikipedia.org/wiki/List_of_Fourier_transforms "List of Fourier transforms") ## References \[[edit](https://en.wikipedia.org/w/index.php?title=List_of_Laplace_transforms&action=edit&section=8 "Edit section: References")\] 1. **[^](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-1)** Distefano, J. J.; Stubberud, A. R.; Williams, I. J. (1995), *Feedback systems and control*, Schaum's outlines (2nd ed.), McGraw-Hill, p. 78, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-07-017052-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-017052-0 "Special:BookSources/978-0-07-017052-0") 2. **[^](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-riley_2-0)** Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010), *Mathematical methods for physics and engineering* (3rd ed.), Cambridge University Press, p. 455, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-521-86153-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-86153-3 "Special:BookSources/978-0-521-86153-3") 3. **[^](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-3)** Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009), "Chapter 33: Laplace transforms", *Mathematical Handbook of Formulas and Tables*, Schaum's Outline Series (3rd ed.), McGraw-Hill, p. 192, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-07-154855-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-154855-7 "Special:BookSources/978-0-07-154855-7") 4. **[^](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-4)** Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009), "Chapter 33: Laplace transforms", *Mathematical Handbook of Formulas and Tables*, Schaum's Outline Series (3rd ed.), McGraw-Hill, p. 183, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-07-154855-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-154855-7 "Special:BookSources/978-0-07-154855-7") 5. **[^](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-5)** ["Laplace Transform"](http://mathworld.wolfram.com/LaplaceTransform.html). *Wolfram MathWorld*. Retrieved 30 April 2016. 6. ^ [***a***](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-bracewell_6-0) [***b***](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-bracewell_6-1) [***c***](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-bracewell_6-2) [***d***](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-bracewell_6-3) Bracewell, Ronald N. (1978), *The Fourier Transform and its Applications* (2nd ed.), McGraw-Hill Kogakusha, p. 227, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-07-007013-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-007013-4 "Special:BookSources/978-0-07-007013-4") 7. ^ [***a***](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-williams_7-0) [***b***](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-williams_7-1) [***c***](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-williams_7-2) [***d***](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-williams_7-3) [***e***](https://en.wikipedia.org/wiki/List_of_Laplace_transforms#cite_ref-williams_7-4) Williams, J. (1973), *Laplace Transforms*, Problem Solvers, George Allen & Unwin, p. 88, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-04-512021-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-04-512021-5 "Special:BookSources/978-0-04-512021-5")  Retrieved from "<https://en.wikipedia.org/w/index.php?title=List_of_Laplace_transforms&oldid=1287808260>" [Categories](https://en.wikipedia.org/wiki/Help:Category "Help:Category"): - [Laplace transforms](https://en.wikipedia.org/wiki/Category:Laplace_transforms "Category:Laplace transforms") - [Mathematics-related lists](https://en.wikipedia.org/wiki/Category:Mathematics-related_lists "Category:Mathematics-related lists") Hidden categories: - [Articles with short description](https://en.wikipedia.org/wiki/Category:Articles_with_short_description "Category:Articles with short description") - [Short description is different from Wikidata](https://en.wikipedia.org/wiki/Category:Short_description_is_different_from_Wikidata "Category:Short description is different from Wikidata") - This page was last edited on 28 April 2025, at 16:09 (UTC). - Text is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://en.wikipedia.org/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License "Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License"); additional terms may apply. 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