Half range Fourier series

Short description: Fourier series defined on an interval [0,L]

A half range Fourier series is a Fourier series defined on an interval $[0,L]$ instead of the more common $[-L,L]$, with the implication that the analyzed function $f(x),x\in [0,L]$ should be extended to $[-L,0]$ as either an even (f(-x)=f(x)) or odd function (f(-x)=-f(x)). This allows the expansion of the function in a series solely of sines (odd) or cosines (even). The choice between odd and even is typically motivated by boundary conditions associated with a differential equation satisfied by $f(x)$.

Example

Calculate the half range Fourier sine series for the function $f(x)=\cos (x)$ where $0<x<\pi$.

Since we are calculating a sine series, $a_n=0\mathrm{\forall }n$ Now, $b_n=\frac{2}{\pi }\underset{0}{\overset{\pi }{\int }}\cos (x)\sin (nx)\mathrm{d}x=\frac{2n((-1{)}^n+1)}{\pi (n^2-1)}\mathrm{\forall }n\ge 2$

When n is odd, $b_n=0$ When n is even, $b_n=\frac{4n}{\pi (n^2-1)}$ thus $b_{2k}=\frac{8k}{\pi (4k^2-1)}$

With the special case $b_1=0$, hence the required Fourier sine series is

$\cos (x)=\frac{8}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{n}{(4n^2-1)}\sin (2nx)$