MATH201/content/Periodic functions (lec 19).md

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#start of lec 19
This lecture we will learn about periodic functions, specifically, non-sinusoidal periodic functions.
# Periodic function
Definition:
$f$ is periodic with period $T \quad (T>0)$ if:
$$f(t)=f(t+T), \quad \forall\ t\in \mathbb{R}$$
![[Drawing 2023-10-20 13.06.35.excalidraw]]
We will now compute laplace transforms of these periodic functions. Computing DE's containing these periodic functions using something like #voparam would not be easy.
If we take the windowed version of the function (one period, where everywhere else is 0, ie:)
$f_{T}(t)=\begin{cases}f(t)\ ,\ & 0\leq t\leq T \\0\ ,\ & \text{otherwise}\end{cases}$
we can "glue together" many of these windows together to rebuild our $f(t)$, like this:
$f(t)=f_{T}(t)+f_{T}(t-T)u(t-T)+f_{T}(t-2T)u(t-2T)+\dots$
$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\}+\mathcal{L}\{f(t-T)u(t-T)\}+\dots$
recall the formula from last lec: $\mathcal{L}\{u(t-a)f(t-a)\}=e^{-as}F(s)$
then:
$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\}(1+e^{-TS}+e^{-2TS}+e^{-3TS}+\dots)$
$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\}(1+e^{-TS}+(e^{-TS})^{2}+(e^{-TS})^{3}+\dots)$
This is a geometric series! $1+r+r^2+\dots$
Geometric series are convergent when $|r|<1$
and equal to: $\frac{1}{1-r}$
in this case, $r=e^{-Ts}$
so:
$$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\} \frac{1}{1-e^{-Ts}}$$
handy formula! ^ will be used again.
#ex
imagine another function: (image is of a square wave with a period of 2a, oscillates between 1 and 0, starts at 1 when t=0.)
![[Drawing 2023-10-20 13.27.58.excalidraw]]
$\mathcal{L}\{f\}=\mathcal{L}\{f_{2a}\} \frac{1}{1-e^{-2as}}$
$f_{2a}=u(t)-u(t-a)$ (this is the first period piece)
$\implies \mathcal{L}\{f_{2a}\}=\mathcal{L}\{u(t)\}-\mathcal{L}\{u(t-a)\}=\frac{1}{s}- \frac{e^{-as}}{s}$
plug back in:
$\mathcal{L}\{f\}=\mathcal{L}\{f_{2a}\} \frac{1}{1-e^{-2as}}=\frac{1}{s}\cancel{ (1-e^{-as}) } \frac{1}{\cancel{ (1-e^{-as}) }(1+e^{-as})}$
$$\mathcal{L}\{f\}=\frac{1}{s(1+e^{-as})}$$
#ex
$y''+3y'+2y=f(t)$ where $f(t)$ is from the previous example
$y(0)=y'(0)=0,\ a=1$ (a is width of 1/2 period in the function f(t))
$s^2Y+3sY+2Y=\mathcal{L}\{f\}= \frac{1}{s(1+e^{-s})}$
$Y(s)=\frac{1}{s(s+1)(s+2)} \frac{1}{1+e^{-s}}$
$=F(s) \frac{1}{1+e^{-s}}$
$\mathcal{L}^{-1}\{F\}=\mathcal{L}^{-1}\{\frac{1}{2} \frac{1}{s}+\frac{1}{2} \frac{1}{s+2}-\frac{1}{s+1}\}$
$=\frac{1}{2}+\frac{1}{2}e^{-2t}-e^{-t}$
$y(t)= \dots$
we want to use formula from earlier
so we need to change $F(s) \frac{1}{1+e^{-s}}$
to: $F(s) \frac{1e^{-s}}{1-e^{-2s}}$
$y(t)=p(t)$, periodic with period of 2 ($T=2$)
$$f_{2a}(t)=\mathcal{L}^{-1}\{F(s)-F(s)e^{-s}\}=\frac{1}{2}+\frac{1}{2}e^{-2t}-e^{-t}-\left( \frac{1}{2}+\frac{1}{2} e^{-2(t-1)}-e^{-(t-1)})u(t-1 \right)$$