Laplace Transform

Definition

Let $f$ be a function on $[0,\mathrm{\infty })$. The Laplace transform of $f$ is the function ${ℒ}\left\{f\right\}$ defined by the integral

${ℒ}\left\{f\right\}(s)=\underset{0}{\overset{\mathrm{\infty }}{\int }}f(t){\mathrm{e}}^{-st}\mathrm{d}t$

The domain of ${ℒ}\left\{f\right\}$ is all the values of $s$ for which the integral exists.

Domain of Existence

Suppose $f(t)\in O\left({\mathrm{e}}^{at}\right)$:

$\begin{array}{rl}|f(t)|& \le k{\mathrm{e}}^{at}\\ |f(t){\mathrm{e}}^{-st}|& \le k{\mathrm{e}}^{(a-s)t}\end{array}$

Then the integral $\underset{0}{\overset{\mathrm{\infty }}{\int }}{\mathrm{e}}^{(a-s)t}\mathrm{d}t$ is convergent for $s>a$

Laplace Transform of Derivatives

${ℒ}\left\{f^{\prime }\right\}(s)=s{ℒ}\left\{f\right\}(s)-f(0)$

Proof

$\begin{array}{rl}{ℒ}\left\{y^{\prime }\right\}& =\underset{0}{\overset{\mathrm{\infty }}{\int }}{y}^{\prime }(t){\mathrm{e}}^{-st}\mathrm{d}t\\ & =\underset{N\to \mathrm{\infty }}{\lim }{y{\mathrm{e}}^{-st}|}_0^N+s\underset{0}{\overset{N}{\int }}{\mathrm{e}}^{-st}y\mathrm{d}t\\ & =\cancel{\underset{N\to \mathrm{\infty }}{\lim }y(N){\mathrm{e}}^{-sN}}-y(0)+s{ℒ}\left\{y\right\}(s)\end{array}$

Therefore

${ℒ}\left\{y^{\prime }\right\}=s{ℒ}\left\{y\right\}(s)-y(0)$

Q.E.D.

In General

${ℒ}\left\{f^{(n)}\right\}(s)=s^n{ℒ}\left\{f\right\}(s)-s^{n-1}f(0)-s^{n-2}{f}^{\prime }(0)-\dots -f^{(n-1)}(0)$

Inverse Laplace Transform

Given $F(s)$ and $f(t)$ such that ${ℒ}\left\{f\right\}(s)=F(s)$, the inverse Laplace transform is defined

$f={{ℒ}}^{-1}\left\{F\right\}$

Laplace Transform of Common Functions

$f(t)$	${ℒ}\left\{f\right\}(s)$
$1$	$\frac{1}{s}$
${\mathrm{e}}^{at}$	$\frac{1}{s-a}s>a$
$t^{n}n\in {\mathbb{\mathbb{Z}}}^{+}$	$\frac{n!}{s^{n+1}}$
$u_c(t)f(t-c)$	${\mathrm{e}}^{-cs}{ℒ}\{f(t)\}$
$\delta (t-c)$	${\mathrm{e}}^{-cs}$
Combinations
${\mathrm{e}}^{at}{t}^n$	$\frac{n!}{(s-a{)}^{n+1}}$
${\mathrm{e}}^{at}\sin bt$	$\frac{b}{(s-a{)}^2+b^2}$
${\mathrm{e}}^{at}\cos bt$	$\frac{s-a}{(s-a{)}^2+b^2}$

Other Properties

$\begin{array}{rl}{ℒ}\{{\mathrm{e}}^{at}f(t)\}(s)& ={ℒ}\{f(t)\}(s-a)\\ {ℒ}\{t^{n}f(t)\}(s)& =(-1{)}^n{F}^{(n)}(s)\end{array}$