Half range Fourier series

In mathematics, a half range Fourier series is a Fourier series defined on an interval ${\displaystyle [0,L]}$ instead of the more common ${\displaystyle [-L,L]}$, with the implication that the analyzed function ${\displaystyle f(x),x\in [0,L]}$ should be extended to ${\displaystyle [-L,0]}$ 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 ${\displaystyle f(x)}$.

Example

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

Since we are calculating a sine series, ${\displaystyle a_{n}=0\ \quad \forall n}$ Now, ${\displaystyle b_{n}={\frac {2}{\pi }}\int _{0}^{\pi }\cos(x)\sin(nx)\,\mathrm {d} x={\frac {2n((-1)^{n}+1)}{\pi (n^{2}-1)}}\quad \forall n\geq 2}$

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

With the special case ${\displaystyle b_{1}=0}$, hence the required Fourier sine series is

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