Half range Fourier series

In mathematics, 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, $b_{n}={\frac {2}{\pi }}\int _{0}^{\pi }\cos(x)\sin(nx)\,\mathrm {d} x={\frac {2n((-1)^{n}+1)}{\pi (n^{2}-1)}}\quad \forall n\geq 2$

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

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