Fourier Series

The Fourier series is a trigonometric series used to analyse periodic functions, which arise frequently in physical applications whenever repetitive or oscillatory phenomena occur. It is a representation of a function, say $f(x)$, made up of sine and/or cosine functions of different frequencies. Examples of applications include:

• The voltage in an alternating circuit,

where $A, \omega$ and $\phi$ are constants denoting the amplitude, angular frequency and phase angle, respectively. This function has period $T=\frac{2 \pi}{\omega}$.

• A full wave rectifier which produces an output violtage equivalent to:

This function has period $T=\frac{\pi}{\omega}$.

• A more complicated example of a periodic function may be the air pressure in a sound wave at a fixed point in space (applications in music & sound).

In this course we will mostly use Fourier series to solve certain linear partial differential equations (see Topic B2). In particular, Fourier series are used to solve boundary-value problems which consist of an ordinary differential equation (ODE) together with a set of boundary conditions (two for a second order ODE) where the conditions are given at different points of the independent variable. Further, the conditions may define the function or its derivative or both.

We start off by giving a definition of the Fourier series expansion of a function, $f(x)$.

Definition 1.1 Assume $f(x)$ is periodic with period $T=2 \pi$. Then, it is possible to express $f(x)$ in terms of a trigonometric series:

where $n$ is an integer. Equation (1.3) is referred to as the Fourier series of $f(x)$ if $a_{0}, a_{n}, b_{n}$ are the Fourier Coefficients defined by formulae whose form is discussed later. Note that a function $f(x)$ is periodic with period $T$ if:

for all values of $x$.