PX275 - H1 - complex fourier series

PX153 - J1 - introduction

the fourier series of a function that is periodic over $2 L :$

f (x) = \frac{A_{0}}{2} + \sum_{n = 1}^{\infty} [A_{n} \cos (\frac{n π x}{L}) + B_{n} \sin (\frac{n π x}{L})]

for periodic over $L :$

f (x) = \frac{A_{0}}{2} + \sum_{n = 1}^{\infty} [A_{n} \cos (\frac{2 n π x}{L}) + B_{n} \sin (\frac{2 n π x}{L})]

using complex exponential notation:

\begin{matrix} \cos θ = \frac{e^{i θ} + e^{- i θ}}{2} \\ \sin θ = \frac{e^{i θ} - e^{- i θ}}{2 i} \end{matrix}

the fourier series can be written as:

\begin{matrix} f (x) = \frac{A_{0}}{2} + \sum_{n = 1}^{\infty} [A_{n} \frac{1}{2} (e^{i 2 π n x / L} + e^{i 2 π n x / L}) + B_{n} \frac{1}{2 i} (e^{i 2 π n x / L} - e^{i 2 π n x / L})] \\ f (x) = \frac{A_{0}}{2} + \sum_{n = 1}^{\infty} [A_{n} \frac{1}{2} (e^{i k x} + e^{i k x}) + B_{n} \frac{1}{2 i} (e^{i k x} - e^{i k x})] \\ = \frac{A_{0}}{2} + \sum_{n = 1}^{\infty} [e^{i k x} (\frac{A_{n}}{2} - \frac{i B_{n}}{2}) + e^{- i k x} (\frac{A_{n}}{2} + \frac{i B_{n}}{2})] \end{matrix}

the bracketed terms are complex coefficients, calling them $C_{n}$ , and exploiting the symmetry:

f (x) = \sum_{n = - \infty}^{\infty} C_{n} e^{i k x}

\begin{matrix} C_{0} = \frac{A_{0}}{2} & n = 0 \\ C_{n} = \frac{A_{n} - i B_{n}}{2} & n \geq 1 \\ C_{- n} = \frac{A_{n} + i B_{n}}{2} & n \leq - 1 \end{matrix}

this is the complex fourier series
in physics, often only concerned with the real part
if $f (x)$ is real valued, ie: $f (x) = f^{*} (x) :$

\begin{matrix} f (x) = \sum_{n = - \infty}^{\infty} C_{n} e^{i k x} \\ f^{*} (x) = \sum_{n = - \infty}^{\infty} C_{n}^{*} e^{- i k x} \end{matrix}

this requires:

\begin{matrix} C_{n}^{*} = C_{- n} \\ C_{- n}^{*} = C_{n} \end{matrix}