14.1. Fourier Series#
For a \(T\)-periodic function,
\[g(t) \sim \frac{a_0}{2} + \sum_{n=1}^{+\infty} \left[ a_n \, \cos\left( n \frac{2\pi}{T} t \right) + b_n \, \sin\left( n \frac{2 \pi }{T} t \right) \right] \ ,\]
todo Prove it with properties of integrals of \(\sin\) and \(\cos\) over \(t \in \left[ 0, T \right]\); prove convergence to average value at jumps
The exponential form reads
(14.1)#\[g(t) \sim \sum_{n=-\infty}^{+\infty} c_n e^{i n \frac{2 \pi }{T}t} \ ,\]
where
(14.2)#\[c_n = \frac{1}{T} \int_{t=0}^{T} f(t) \, e^{-i n \frac{2\pi}{T} t} \ .\]
Proof
Exploiting the properties of integrals of complex exponentials with \(k \in \mathbb{Z}\)
\[\begin{split}\int_{t=0}^{T} e^{i k \frac{2 \pi}{T} t} \, dt =
\begin{cases}
\frac{1}{i k \frac{2 \pi}{T}} \left.\left[ e^{ik \frac{2\pi}{T} t} \right]\right|_{t=0}^{T} = 0 & \hfill \qquad \text{if $ k \ne 0$} \\
T & \hfill \text{if $ k = 0$}
\end{cases}
\end{split}\]
\[\int_{t=0}^{T} f(t) e^{-i m \frac{2 \pi}{T} t} \, dt
\sim \int_{t=0}^{T} \sum_{n=-\infty}^{+\infty} c_n e^{i n \frac{2 \pi }{T}t} e^{-i m \frac{2 \pi}{T} t}
\sim \sum_{n=-\infty}^{+\infty} c_n \sim \int_{t=0}^{T} e^{i (n-m) \frac{2 \pi }{T}t}
\sim T \, c_m \ .
\]