Laplace Transform

A transform that is useful in analyzing linear dynamical systems because

• differentiation becomes multiplication by $s$
• integration becomes division by $s$

Definition

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

Existence

If

• $f(t)$ is piecewise continuous (continuous except for a finite number of jump/hole discontinuities) on $[0,\infty )$
• $f(t)\in O(e^{\alpha t})$ (Exponential Order from Big O, Omega, Theta Notation)
  • If ${\lim }_{t\to \infty }\frac{f(t)}{e^{\alpha t}}=0$ Then $\mathcal{L}\{f(t)\}$ exists for $s>\alpha$

Note: these are sufficient but not necessary conditions (there are transforms that exist but do not satisfy these)

Properties

• ${\lim }_{s\to 0}F(s)=0$
• $\mathcal{L}\{c_1{f}_1+c_2{f}_2\}=c_1\mathcal{L}\{f_1\}+c_2\mathcal{L}\{f_2\}$

Inverse

• Go the opposite of the table
• Often needs Partial Fraction Decomposition

Table of Transforms

$f(t)$	$F(s)$
1	$\frac{1}{s},s>0$
$e^{at}$	$\frac{1}{s-a},s>a$
$t^n(n>0)$	$\frac{n!}{s^{n+1}},s>0$
$\sin \beta t$	$\frac{b}{s^2+b^2},s>0$
$\cos \beta t$	$\frac{s}{s^2+b^2},s>0$
$t\sin \beta t$	$\frac{2bs}{(s^2+b^2{)}^2}s>0$
$t\cos \beta t$	$\frac{s^2-b^2}{(s^2+b^2{)}^2}s>0$
$e^{\alpha t}f(t)$	$F(s-a),s>a$
$f^{\prime }(t)$	$sF(s)-f(0)$
$f^{\prime \prime }(t)$	$s^{2}F(s)-sf(0)-f^{\prime }(0)$
$tf(t)$	$-\frac{dF}{ds}$
$t^{n}f$	$(-1{)}^n\frac{d^n}{d{s}^n}(\mathcal{L}\{F\}(s))$

Things to Use

• Discontinuous functions: Unit Step Function
• Multiplication: Convolution
• Equal to unspecified function: Impulse Response Function
• Dirac Delta Function