The Fourier series of a 2L-periodic function \(f(x)\) is defined as:

\(\displaystyle f(x)=\frac{a_{0}}{2}+\sum\limits^{\infty}_{n=1}a_{n}\cos\left(\frac{n\pi x}{L}\right)+b_{n}\sin\left(\frac{n\pi x}{L}\right)\)

where
- \(\displaystyle a_{0}=\frac{1}{L}\int^{L}_{-L}f(x)dx\equiv \left\langle1,f\right\rangle\).
- \(\displaystyle a_{n}=\frac{1}{L}\int^{L}_{-L}f(x)\cos\left(\frac{n\pi x}{L}\right)dx\equiv \frac{\left\langle\cos_{n},f\right\rangle}{\left\langle\cos_{n},\cos_{n}\right\rangle}\).
- \(\displaystyle b_{n}=\frac{1}{L}\int^{L}_{-L}f(x)\sin\left(\frac{n\pi x}{L}\right)dx\equiv \frac{\left\langle\sin_{n},f\right\rangle}{\left\langle\sin_{n},\sin_{n}\right\rangle}\).