ece205: orthogonality and fourier

docs/2a/ece205.md

@@ -245,6 +245,37 @@ If functions $f$ and $g$ have a period $T$, then both $af+bg$ and $fg$ also must
 - odd × odd = even
 - even × odd = odd
 
+## Orthogonality
+
+$$\int^L_{-L}\cos(\frac{m\pi x}{L})\sin(\frac{n\pi x}{L})dx=0$$
+
+$$
+\int^L_{-L}\cos(\frac{m\pi x}{L})(\frac{n\pi x}{L})dx=\begin{cases}
+2L & \text{if }m=n=0 \\
+L & \text{if }m=n\neq 0 \\
+0 & \text{if }m\neq n
+\end{cases}
+$$
+
+$$
+\int^L_{-L}\sin(\frac{m\pi x}{L}\sin(\frac{n\pi x}{L})dx=\begin{cases}
+L & \text{if }m=n \\
+0 & \text{if }m\neq n
+\end{cases}
+$$
+
+Functions are **orthogonal** on an interval when the integral of their product is zero, and a set of functions is **mutually orthogonal** if all functions in the set are orthogonal to each other.
+
+If a Fourier series converges to $f(x)$:
+
+$$f(x)=\frac{a_0}{2} + \sum^\infty_{n=1}\left(a_n\cos(\frac{n\pi x}{L})+b_n\sin(\frac{n\pi x}{L})\right)$$
+
+The **Euler-Fourier** formulae must apply:
+$$
+\boxed{a_n=\frac 1 L\int^L_{-L}f(x)\cos(\frac{n\pi x}{L})dx} \\
+\\
+\boxed{b_n=\frac 1 L\int^L_{-L}f(x)\sin(\frac{n\pi x}{L})dx}
+$$
 
 ## Resources