Laplace Transform

Definition

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

${\mathcal {L}}\left\{f\right\}(s)=\int _{0}^{\infty }f(t)\,\mathrm {e} ^{-s\,t}\,\mathrm {d} t$

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

Domain of Existence

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

${\begin{aligned}\left|f(t)\right|&\leq k\mathrm {e} ^{a\,t}\\\left|f(t)\,\mathrm {e} ^{-s\,t}\right|&\leq k\,\mathrm {e} ^{(a-s)t}\end{aligned}}$

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

Laplace Transform of Derivatives

${\mathcal {L}}\left\{f'\right\}(s)=s\,{\mathcal {L}}\left\{f\right\}(s)-f(0)$

Proof

${\begin{aligned}{\mathcal {L}}\left\{y'\right\}&=\int _{0}^{\infty }y'(t)\,\mathrm {e} ^{-s\,t}\,\mathrm {d} t\\&=\lim _{N\to \infty }\left.y\,\mathrm {e} ^{-s\,t}\right|_{0}^{N}+s\,\int _{0}^{N}\mathrm {e} ^{-s\,t}y\,\mathrm {d} t\\&={\cancel {\lim _{N\to \infty }y(N)\,\mathrm {e} ^{-s\,N}}}-y(0)+s\,{\mathcal {L}}\left\{y\right\}(s)\end{aligned}}$

Therefore ${\mathcal {L}}\left\{y'\right\}=s\,{\mathcal {L}}\left\{y\right\}(s)-y(0)$

Q.E.D.

In General

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

Inverse Laplace Transform

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

$f={\mathcal {L}}^{-1}\left\{F\right\}$

Laplace Transform of Common Functions

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

Other Properties

${\begin{aligned}{\mathcal {L}}\left\{\mathrm {e} ^{a\,t}\,f(t)\right\}(s)&={\mathcal {L}}\left\{f(t)\right\}(s-a)\\{\mathcal {L}}\left\{t^{n}\,f(t)\right\}(s)&=(-1)^{n}\,F^{(n)}(s)\end{aligned}}$