Laplace Transform

Abstract

The purpose of the Laplace Transform is to transform functions and equations that exist in the time-domain into the more complex frequency-domain. This is done for numerous reason, most of which primarily being that there are analysis techniques that can only be applied in the frequency domain, and generally the frequency-domain is more easy to conduct certain mathematical techniques inside.

Formal Definition

$$F(s)={\int }_0^{\infty }f(t)e^{-st}dt$$

Example

Consider F(t) = 1

$$F(s)={\int }_0^{\infty }{e}^{-st}dt=\frac{-1}{s}[e^{-st}{]}_0^{\infty }$$

This is equivalent to:

$$F(s)=\frac{-1}{s}[0-1]=\frac{1}{s}$$

Common Transforms:

GENERAL

Time	Frequency
$\frac{df}{dt}$	$sF(s)-f(0)$
$\frac{d^{2}f}{d{t}^2}$	$sF(s)-sf(0)-\frac{df}{dt}(0)$
$\frac{d^{n}f}{d{t}^n}$	$s^{n}F(s)-s^{n-1}f(0)-s^{n-2}\frac{df}{dt}(0)-\frac{d^{n-1}f}{d{t}^{n-1}}(0)$
$f(\alpha t),\alpha >0$	$\frac{1}{\alpha }F(s/\alpha )$
$e^{at}f(t)$	$F(s-a)$
$tf(t)$	$-\frac{dF}{ds}$
$t^{k}f(t)$	$(-1{)}^k\frac{d^{k}F(s)}{d{s}^k}$
$\frac{f(t)}{t}$	${\int }_s^{\infty }F(s)ds$
$t^n$	$\frac{n!}{s^{n+1}}$
*SPECIFIC

Time	Frequency
$1$ (Unit Step)	$\frac{1}{s}$
$\delta$ (Unit Impulse)	1
${\delta }^{(k)}$	$s^k$
t	$\frac{1}{s^2}$
$e^{at}$	$\frac{1}{s-a}$
$\cos \omega t$	$\frac{s}{s^2+{\omega }^2}$
$\sin \omega t$	$\frac{\omega }{s^2+{\omega }^2}$
$e^{-at}\cos \omega t$	$\frac{s+a}{(s+a{)}^2+{\omega }^2}$
$e^{-at}\sin \omega t$	$\frac{\omega }{(s+a{)}^2+{\omega }^2}$