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. | ||
+ | |||
+ | For suitable functions f, the Laplace transform is the integral | ||
+ | |||
+ | <math>\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} \, dt.</math> | ||
+ | |||
{| class="wikitable" | {| class="wikitable" | ||
− | |+ | + | |+ List of Common Laplace Transforms |
|- | |- | ||
− | ! <math> f(t) </math> !! <math> F(s) = \mathcal{L} | + | ! <math> f(t) </math> !! <math> F(s) = \mathcal{L}\{f\}(s) </math> |
|- | |- | ||
| <math> 1 </math> || <math> \frac{1}{s} </math> <math> s > 0 </math> | | <math> 1 </math> || <math> \frac{1}{s} </math> <math> s > 0 </math> | ||
|- | |- | ||
− | | <math> | + | | <math> t </math> || <math> \frac{1}{s^2} </math> <math> s > 0 </math> |
|- | |- | ||
− | | <math> t </math> || <math> \frac{ | + | | <math> t^{n} </math> || <math> \frac{n!}{s^{n+1}} </math> <math> s > 0 </math> |
|- | |- | ||
− | | <math> t^{n} </math> || <math> \frac{n!}{s^{n+1}} </math> | + | | <math> t^{n}e^{at} </math> || <math> \frac{n!}{s^{n+1}} </math> <math> s > a </math> |
|- | |- | ||
− | | <math> | + | | <math> e^{at} </math> || <math> \frac{1}{s-a} </math> <math> s > a </math> |
|- | |- | ||
− | | <math> \sin{at} </math> || <math> \frac{a}{s^2+a^2} </math> | + | | <math> \sin{at} </math> || <math> \frac{a}{s^2+a^2} </math> <math> s > 0 </math> |
|- | |- | ||
− | | <math> \cos{at} </math> || <math> \frac{s}{s^2+a^2} </math> | + | | <math> \cos{at} </math> || <math> \frac{s}{s^2+a^2} </math> <math> s > 0 </math> |
|- | |- | ||
− | | <math> \sinh{at} </math> || <math> \frac{a}{s^2-a^2} </math> | + | | <math> \sinh{at} </math> || <math> \frac{a}{s^2-a^2} </math> <math> s > |a| </math> |
|- | |- | ||
− | | <math> \cosh{at} </math> || <math> \frac{s}{s^2-a^2} </math> | + | | <math> \cosh{at} </math> || <math> \frac{s}{s^2-a^2} </math> <math> s > |a| </math> |
|- | |- | ||
− | | <math> e^{at}\sin{bt} </math> || <math> \frac{b}{(s-a)^2+b^2} </math> | + | | <math> e^{at}\sin{bt} </math> || <math> \frac{b}{(s-a)^2+b^2} </math> <math> s > a </math> |
|- | |- | ||
− | | <math> e^{at}\cos{bt} </math> || <math> \frac{s-a}{(s-a)^2+b^2} </math> | + | | <math> e^{at}\cos{bt} </math> || <math> \frac{s-a}{(s-a)^2+b^2} </math> <math> s > a </math> |
|- | |- | ||
− | | <math> e^{at}\sinh{bt} </math> || <math> \frac{b}{(s-a)^2-b^2} </math> | + | | <math> e^{at}\sinh{bt} </math> || <math> \frac{b}{(s-a)^2-b^2} </math> <math> (s - a) > |b| </math> |
|- | |- | ||
− | | <math> e^{at}\cosh{bt} </math> || <math> \frac{s-a}{(s-a)^2-b^2} </math> | + | | <math> e^{at}\cosh{bt} </math> || <math> \frac{s-a}{(s-a)^2-b^2} </math> <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 | ||
+ | |||
+ | == Licensing == | ||
+ | Content obtained and/or adapted from: | ||
+ | * [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
Resources
- Laplace Transforms, Paul's Online Notes
Licensing
Content obtained and/or adapted from:
- Laplace transform, Wikipedia under a CC BY-SA license