Conditions for Fourier Series:

• The function is periodic
• we only care about the function in a finite range

🗒️ Note: if the function is effectively infinite We can use a Fourier transform

Simple function Fourier series

Given a function $f(x)$ with period $2L$ in the range $-L$ to $+L$ then we can perform the following expansion:

$$ f(x)=\frac{a_0}{2}+\sum^\infin_{n=1}\left [ a_n \cos\left ( \frac{n\pi x}{L}\right ) + b_n \cos\left ( \frac{n\pi x}{L}\right ) \right ] $$

where

$$ a_n=\frac{1}{L} \int^{+L}{-L} f(x) \cos\left ( \frac{n \pi x}{L} \right ) \,\text dx \qquad b_n=\frac{1}{L}\int^{+L}{-L} f(x) \,\sin\left ( \frac{n \pi x}{L}\right )\,\text dx $$

Dirichlet’s conditions (the conditions for this expansion)

• Single values (the function needs to only have 1 output $x$)
• Finite number of discontinuities
• $\int^{+L}_{-L}|f(x)|\,\text dx$ must be finite

This can be used to represent any function

Orthogonal properties of Fourier series

the basis functions ($\sin$ and $\cos$) are orthogonal

$$ \begin{aligned} \int^{+L}{-L}\sin\left ( \frac{n \pi x}{L}\right )\cos \left ( \frac{m \pi x}{L}\right )\,\text dx&=0 \\ \int^{+L}{-L}\sin \left ( \frac{n \pi x}{L}\right ) \sin \left ( \frac{m \pi x}{L}\right )\,\text dx&=L\delta_{nm} \\ \int^{+L}_{-L}\cos\left ( \frac{n \pi x}{L}\right )\cos\left ( \frac{m \pi x}{L}\right )\,\text dx&=\left \{ \begin{matrix} 0 & \text{if} & m\ne n \\ L & \text{if} & m=n \\ 2L & \text{if} & m=n=0 \end{matrix} \right .

\end{aligned} $$

the Kronecker delta is defines as

$$ \delta_{nm}=\left \{ \begin{matrix} 1 & m=n \\ 0& m\ne n \end{matrix}\right . $$

any 2 functions is orthogonal on an interval $a \le x \le b$ if

$$ \int^b_a u(x) v(x)\,\text dx=0 $$