ece205: add fourier

docs/2a/ece205.md

@@ -277,6 +277,37 @@ $$
 \boxed{b_n=\frac 1 L\int^L_{-L}f(x)\sin(\frac{n\pi x}{L})dx}
 $$
 
+!!! example
+    The Fourier series for the square wave function: $f(x)=\begin{cases}-1 & -\pi < x < 0 \\ 1 & 0 < x < \pi\end{cases}$
+    
+    The period is clearly $2\pi\implies L=\pi$. $f(x)$ is also odd, by inspection.
+    
+    \begin{align*}
+    a_n&=\frac 1\pi\int^\pi_{-\pi}\underbrace{f(x)\cos(\frac{n\pi x}{\pi})}_\text{odd × even = odd}dx=0=a_0 \\
+    b_n&=\frac 1 \pi\int^\pi_{-\pi}f(x)\sin(\frac{n\pi x}{\pi})dx \\
+    \tag{even}&=\frac 2\pi\int^\pi_0f(x)\sin(nx)dx \\
+    \tag{$f(x)>1$ when $x>0$}&=\frac 2\pi\int^\pi_0\sin(nx)dx \\
+    &=\frac 2\pi\left[\frac{-\cos nx}{n}\right]^\pi_0 \\
+    &=\begin{cases}
+    \frac{4}{\pi n} & \text{if $n$ is odd} \\
+    0 & \text{else}
+    \end{cases}
+    \therefore f(x)&=\sum^\infty_{n=1}\frac 2\pi\left(\frac{1-(-1)^n}{n}\sin(nx)\right) \\
+    \tag{only odd $n$s are non-zero}&=\frac4\pi\sum^\infty_{n=1}\frac{1}{2n-1}\sin[(2n-1)x]
+    \end{align*}
+    
+    Thus the Fourier series is $$.
+
+### Separation of variables
+
+To solve IBVPs, where $X(x)$ and $T(t)$ are exclusively functions of their respective variables:
+
+$$u(x,t)=X(x)T(t)$$
+
+Substituting it into the IBVP results in a **separation constant** $-\lambda$.
+
+$$\boxed{\frac{T'(t)}{a^2T(t)}=\frac{X''(x)}{X(x)}=-\lambda}$$
+
 ## Resources
 
 - [Laplace Table](/resources/ece/laplace.pdf)