Half range Fourier series

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, $a_n=0\ \quad \forall n$ Now, $b_n= \frac{2}{\pi} \int_0^\pi \cos(x)\sin(nx)\,\mathrm{d}x = \frac{2n((-1)^n+1)}{\pi(n^2-1)}\quad \forall n\ge 2$

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

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

$\cos(x) = {{8 \over \pi} \sum_{n=1}^{\infty} {n \over(4n^2-1)}\sin(2nx)}$