# Laplace Transform

Given a function $f(t)$, the Laplace Transform of $f(t)$ denoted by $F(s)$ (given that the integral exists) is defined by

$F(s)=L{f(t)}=∫_{−∞}f(t)e_{−st}dt$

- $s$ is a complex parameter called the frequency

Why is

`s`

called the frequency?In $s=σ+jω$, the term $ω$ relates to angular frequency in radians per second.

The teacher gives these two examples:

Laplace Transforms of common functions:

### Properties of Laplace Transform

Theorem 1

For any function $f(t)$, the one-sided Laplace transform will always converge for some $s$ that $Re(s)$ is sufficiently large.

Theorem 2: Laplace Transform is Linear

Suppose that $f(t)$ and $g(t)$ have Laplace transforms $F(s)$ and $G(s)$. Then for all $α,β∈C$

$L{αf(t)+βg(t)}=αF(s)+βG(s)$ the the ROC is the intersection of the ROCs for $F(s)$ and $G(s)$.

Theorem 3: Time-Scaling

If $L{f(t)}=F(s)$ then for $c>0,L{f(ct)}=c1 F(cs )$

Theorem 4: Exponential Modulation

$L{e_{αt}f(t)}=F(s−α)$

Theorem 5: Time-Shifting

If $F(s)=L{f(t)u(t)}$ and $g(t)=f(t−T)u(t−T)$ then

$G(s)=e_{−sT}F(s)$

Theorem 6: Multiplication by

`t`

If $L{f(t)}=F(s)$ then $L{tf(t)}=−dsd F(s)$

Theorem 7: Laplace Transform of a Derivative/Integral

Let $f(t)$ be such that there is a real value α such that the integral $∫_{0_{−}}∣f(t)∣e_{−αt}dt$

converge and such that there exists a function $f_{′}(t)$ such that for $t≥0$ $f(t)=f(0_{−})+∫_{0_{−}}f_{′}(τ)dτ$

andthere exists a real value $β$ such that $∫_{0_{−}}∣f_{′}(t)∣e_{−βt}dt$converges. In this case $F(s)=s1 f(0_{−})+s1 L{f_{′}(t)}$ or in other-words $L{f_{′}(t)}=sF(s)−f(0_{−})$

- We can use this theorem to solve linear ODEs!