Fourier series

A Fourier series is an expansion of periodic function into an infinite sum of sines or cosines. In general, any well behaved periodic function can be written as a Fourier series.

Consider a function $f(x)$ with period $L$ . We can write it as an infinite sum

f(x) = \sum_{m=1}^\infty A_m \sin\left(\frac{m\pi x}{L}\right). \tag{1}

We use the orthogonality of the sine and cosine functions, which states

\int_0^L \sin\left(\frac{m\pi x}{L}\right) \sin\left(\frac{n\pi x}{L}\right) dx = \begin{cases} \frac{L}{2} & \text{if } m=n \\ 0 & \text{otherwise}. \end{cases}

(the above holds for $\cos$ as well).

We multiply both sides of (1) by $\sin\left(\frac{n\pi x}{L}\right)$ for some arbitrary integer $n$ and integrate.

\begin{align*} \int_0^L \sin\left(\frac{n\pi x}{L}\right) f(x) dx &= \int_0^L \sum_m^\infty A_m \sin\left(\frac{m\pi x}{L}\right) \sin\left(\frac{n\pi x}{L}\right) dx. \end{align*}

We can change the order of the sum and integral on the right. We then notice that for all $m \ne n$ , the integral is zero; so only the $m=n$ term of the sum remains. By orthogonality we know the value of the integral.

\begin{align*} \int_0^L \sin\left(\frac{n\pi x}{L}\right) f(x) dx &= \frac{A_n L}{2} \\ \frac{2}{L} \int_0^L \sin\left(\frac{n\pi x}{L}\right) f(x) dx &= A_n. \end{align*}

We have found the amplitudes. Plugging in to (1) and setting $n=m$ we get our Fourier series.

f(x) = \sum_{m=1}^\infty \frac{2}{L} \sin\left(\frac{m\pi x}{L}\right) \int_{x'=0}^L \sin\left(\frac{m\pi x'}{L}\right) f(x') dx'.

Note we write the integration variable as $x'$ to distinguish it from the parameter $f(x)$ . The prime does not denote a derivative here.