Difference between revisions of "Laplace Transform"

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In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace, is an integral transform that converts a function of a real variable t (often time) to a function of a complex variable s (complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.
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For suitable functions f, the Laplace transform is the integral
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<math>\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} \, dt.</math>
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{| class="wikitable"
 
{| class="wikitable"
|+ Caption text
+
|+ List of Common Laplace Transforms
 
|-
 
|-
! <math> f(t) </math> !! <math> F(s) = \mathcal{L}|f(t)| </math>
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! <math> f(t) </math> !! <math> F(s) = \mathcal{L}\{f\}(s) </math>
 
|-
 
|-
| <math> 1 </math> || <math> \frac{1}{s} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; s > 0 </math>
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| <math> 1 </math> || <math> \frac{1}{s} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> s > 0 </math>
 
|-
 
|-
| <math> e^{at} </math> || <math> \frac{1}{s-a} </math>
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| <math> t </math> || <math> \frac{1}{s^2} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> s > 0 </math>
 
|-
 
|-
| <math> t </math> || <math> \frac{1}{s^2} </math>
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| <math> t^{n} </math> || <math> \frac{n!}{s^{n+1}} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> s > 0 </math>
 
|-
 
|-
| <math> t^{n} </math> || <math> \frac{n!}{s^{n+1}} </math>
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| <math> t^{n}e^{at} </math> || <math> \frac{n!}{s^{n+1}} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> s > a </math>
 
|-
 
|-
| <math> t^{n}e^{at} </math> || <math> \frac{n!}{s^{n+1}} </math>
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| <math> e^{at} </math> || <math> \frac{1}{s-a} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> s > a </math>
 
|-
 
|-
| <math> \sin{at} </math> || <math> \frac{a}{s^2+a^2} </math>
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| <math> \sin{at} </math> || <math> \frac{a}{s^2+a^2} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> s > 0 </math>
 
|-
 
|-
| <math> \cos{at} </math> || <math> \frac{s}{s^2+a^2} </math>
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| <math> \cos{at} </math> || <math> \frac{s}{s^2+a^2} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> s > 0 </math>
 
|-
 
|-
| <math> \sinh{at} </math> || <math> \frac{a}{s^2-a^2} </math>
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| <math> \sinh{at} </math> || <math> \frac{a}{s^2-a^2} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> s > |a| </math>
 
|-
 
|-
| <math> \cosh{at} </math> || <math> \frac{s}{s^2-a^2} </math>
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| <math> \cosh{at} </math> || <math> \frac{s}{s^2-a^2} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> s > |a| </math>
 
|-
 
|-
| <math> e^{at}\sin{bt} </math> || <math> \frac{b}{(s-a)^2+b^2} </math>
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| <math> e^{at}\sin{bt} </math> || <math> \frac{b}{(s-a)^2+b^2} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> s > a </math>
 
|-
 
|-
| <math> e^{at}\cos{bt} </math> || <math> \frac{s-a}{(s-a)^2+b^2} </math>
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| <math> e^{at}\cos{bt} </math> || <math> \frac{s-a}{(s-a)^2+b^2} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> s > a </math>
 
|-
 
|-
| <math> e^{at}\sinh{bt} </math> || <math> \frac{b}{(s-a)^2-b^2} </math>
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| <math> e^{at}\sinh{bt} </math> || <math> \frac{b}{(s-a)^2-b^2} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> (s - a) > |b| </math>
 
|-
 
|-
| <math> e^{at}\cosh{bt} </math> || <math> \frac{s-a}{(s-a)^2-b^2} </math>
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| <math> e^{at}\cosh{bt} </math> || <math> \frac{s-a}{(s-a)^2-b^2} </math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math> (s - a) > |b|</math>
 
|}
 
|}
  
 
==Resources==
 
==Resources==
 
* [https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx Laplace Transforms], Paul's Online Notes
 
* [https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx Laplace Transforms], Paul's Online Notes
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== Licensing ==
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Content obtained and/or adapted from:
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* [https://en.wikipedia.org/wiki/Laplace_transform Laplace transform, Wikipedia] under a CC BY-SA license

Latest revision as of 19:30, 5 November 2021

In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace, is an integral transform that converts a function of a real variable t (often time) to a function of a complex variable s (complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.

For suitable functions f, the Laplace transform is the integral

List of Common Laplace Transforms
         
         
         
         
         
         
         
         
         
         
         
         
         

Resources

Licensing

Content obtained and/or adapted from: