# LaPlace Transforms

• How do we model a resonator when its forcing function is not continuous such as when a bell is struck?

#### Integral Transforms

• K(s,t)
A function known as the kernel of the transform
• α,β
The limits of the transform
• F(s)
The integral transform of f(t)
• $F ⁡ s = ∫ α β K ⁡ s t ⁢ f ⁡ t ⁢ d t$

#### The Laplace Transform

$L ⁡ f ⁡ t = F ⁡ s = ∫ 0 ∞ e - s ⁢ t ⁢ f ⁡ t ⁢ d t$

The laplace transform of a function only exists only if there exist numbers M and C such that: $f ⁡ t =< M ⁢ e C ⁢ t$ For all values of t > 0.

Important Transforms
f(t) F(s)
$\delta \left(t\right)$
(Dirac Delta Function)
$1$
$u\left(t\right)$
(Heviside Step Function)
$\frac{1}{s}$
tn $\frac{n!}{{s}^{n+1}}$
e-at $\frac{1}{s+a}$
sin(ωt) $\frac{\omega }{{s}^{2}+{\omega }^{2}}$
cos(ωt) $\frac{s}{{s}^{2}+{\omega }^{2}}$
Operational Transforms
f(t) F(s)
C f(t) (multiplication by a constant) C F(s)
f(t)+g(t) F(t)+G(t)
$\frac{df\left(t\right)}{dx}$ $sF\left(s\right)-f\left(0\right)$
$\frac{{d}^{2}f\left(t\right)}{{dx}^{2}}$ ${s}^{2}F\left(s\right)-sf\left(0\right)-\frac{df\left(0\right)}{dx}$
$\frac{{d}^{3}f\left(t\right)}{{dx}^{3}}$ ${s}^{3}F\left(s\right)-{s}^{2}f\left(0\right)-s\frac{df\left(0\right)}{dx}-\frac{{d}^{2}f\left(0\right)}{{dx}^{2}}$
f(t-a)u(t-a), a>0 (time shift) ${e}^{-as}F\left(s\right)$
f(at), a>0 (time scale) $\frac{1}{a}F\left(\frac{s}{a}\right)$

Integral Transforms

The Laplace Transform is an integral transform. It is useful for solving differential equations with non-continuous forcing functions such as modeling the response of a resonator after applying some pulse. s can be a complex value.

Shifts in the time domain

Shifts in the frequency domain

### Solving Differential Equations with Laplace Transforms

Transforming a differential equation into the s domain by means of the laplace transform makes its solution an algebraic problem. Then the solution must be transformed back into the t domain.