Fourier Series

More details about Fourier series can be found in Kreyszig, Chapter 10. Here we simply list some of the important results.

Firstly, if a function $f(x)$, defined over an interval $-L < x < L$, is extended as a periodic function so that

$$f(x+2L) = f(x),$$

then $f(x)$ may be expressed in terms of a Fourier series as

$$f(x) = a_{0} + \sum_{n=1}^{\infty}\left ( a_{n}\cos {n\pi \over L}x + b_{n}\sin {n\pi \over L}x \right ),$$ (2.11)

with the Fourier coefficients of $f(x)$ given by the Euler fromulae

$$a_{0} = {1\over 2L}\int_{-L}^{L}f(x) dx,$$ (2.12)

$$a_{n} = {1\over L}\int_{-L}^{L}f(x) \cos {n\pi\over L}x dx,$$ (2.13)

$$b_{n} = {1\over L}\int_{-L}^{L}f(x) \sin {n\pi \over L}x dx,$$ (2.14)

for positive integer values of $n$. The derivation of theses coefficients requires some knowledge about the orthogonality properties of the trigonometric functions. The main properties that we need are given by the following integrals,

$$\begin{eqnarray*} \int_{-L}^{L}\sin {m\pi \over L}x dx, & = & 0 \\ \int_{-L}^... ... \int_{-L}^{L}\cos^{2} {m\pi \over L}x dx \hbox{ }= \hbox{ }L. \end{eqnarray*}$$

The proof of these integrals all follows the same pattern and requires the use of some trigonometric identities. For example, we may express

$$\int_{-L}^{L}\cos {m\pi \over L}x \cos {n\pi \over L}x dx =... ...x dx + {1\over 2}\int_{-L}^{L}\cos {(m - n)\pi \over L}x dx.$$

The integrand of the first term on the right hand side is always periodic over $-L < x < L$ and so the integral will be zero. The second term on the right hand side is also periodic and will be zero provided that $m \ne n$. The case $m = n$ gives the integral as

$${1\over 2}\int_{-L}^{L}\cos {(m - n)\pi \over L}x dx = {1\over 2}\int_{-L}^{L} dx = L.$$

The other integrals are done in a similar manner, using

$$\begin{eqnarray*} 2 \sin mx \sin nx & = & \cos (m-n)x - \cos (m+n)x, \\ 2 \cos mx \sin nx & = & \sin (m+n)x - \sin (m-n)x. \end{eqnarray*}$$