General Fouries series

So far we have considered periodic functions of period $2\pi$ and their Fourier series. Similar formulae can be obtained for a piecewise continuous periodic function $f$ of an arbitrary period $p$ ($p > 0$). We consider the half-period $L$, such that $p = 2L$. Then we define the function $g(u) = f(\displaystyle{\frac{L}{\pi} u})$, which is also periodic and has the period $2\pi$:

$$\displaystyle{g(u + 2\pi) = f(\frac{L}{\pi}(u + 2\pi)) = f(\frac{L}{\pi} u + 2L) = f(\frac{L}{\pi} u + p) = f(\frac{L}{\pi} u) = g(u).}$$ (10)

Assume that $g$ can be written as

$$g(u) = \frac{a_0}{2} + \sum_{n=1}^{\infty}(a_n \cos nu + b_n \sin nu),$$ (11)

with $a_0 = \displaystyle{\frac{1}{\pi} \int_{-\pi}^{\pi}} g(u) du, \ a_n = \... ... du, \\ b_n = \frac{1}{\pi} \displaystyle{\int_{-\pi}^{\pi}} g(u)\sin nu du.$

Then we change variables by setting $x = \displaystyle{\frac{L}{\pi} u}$, and therefore $u = \displaystyle{\frac{\pi}{L} x}$,
so $du = \displaystyle{\frac{\pi}{L} dx}$, and

$$f(x) = g(u) = g(\displaystyle{\frac{\pi}{L} x}) = \frac{a_0}{2} + \sum_{n=1}^{\infty}(a_n \cos \frac{n\pi x}{L} + b_n \sin \frac{n\pi x}{L}).$$ (12)

The Fourier coefficient $a_n$ is calculated as follows:

$$a_n = \frac{1}{\pi} \displaystyle{\int_{-\pi}^{\pi} g(u)\cos nu \... ...\pi x}{L} dx = \frac{1}{L} \int_{-L}^L f(x) \cos \frac{n\pi x}{L} dx .}$$ (13)

Similarly,

$$b_n = \displaystyle{\frac{1}{L} \int_{-L}^L f(x) \sin \frac{n\pi x}{L} dx ,}$$ (14)

$$a_0 = \displaystyle{\frac{1}{L} \int_{-L}^L f(x) dx .}$$ (15)