Laplace Transform

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

${\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.$

List of Common Laplace Transforms
$f(t)$	$F(s)={\mathcal {L}}\{f\}(s)$
$1$	${\frac {1}{s}}$ $s>0$
$t$	${\frac {1}{s^{2}}}$ $s>0$
$t^{n}$	${\frac {n!}{s^{n+1}}}$ $s>0$
$t^{n}e^{at}$	${\frac {n!}{s^{n+1}}}$ $s>a$
$e^{at}$	${\frac {1}{s-a}}$ $s>a$
$\sin {at}$	${\frac {a}{s^{2}+a^{2}}}$ $s>0$
$\cos {at}$	${\frac {s}{s^{2}+a^{2}}}$ $s>0$
$\sinh {at}$	${\frac {a}{s^{2}-a^{2}}}$ $s>\|a\|$
$\cosh {at}$	${\frac {s}{s^{2}-a^{2}}}$ $s>\|a\|$
$e^{at}\sin {bt}$	${\frac {b}{(s-a)^{2}+b^{2}}}$ $s>a$
$e^{at}\cos {bt}$	${\frac {s-a}{(s-a)^{2}+b^{2}}}$ $s>a$
$e^{at}\sinh {bt}$	${\frac {b}{(s-a)^{2}-b^{2}}}$ $(s-a)>\|b\|$
$e^{at}\cosh {bt}$	${\frac {s-a}{(s-a)^{2}-b^{2}}}$ $(s-a)>\|b\|$