Complex Fourier series and Fourier transform

5.13 Complex Fourier series and Fourier transform

Given $f (x)$ which is periodic on $0 < x < L$ , so period is $L$ , then Fourier series is $f (x) \sim \frac{1}{\sqrt{L}} \sum_{n = - \infty}^{\infty} c_{n} e^{i n \frac{2 π}{L} x}$ Where $\begin{aligned} c_{n} & = ⟨ n | f ⟩ \\ = \frac{1}{\sqrt{L}} \int_{0}^{L} f (x) e^{- i n \frac{2 π}{L} x} d x \end{aligned}$

The basis are $| n ⟩ = \frac{1}{\sqrt{L}} e^{- i n \frac{2 π}{L} x}$ and $L$ is the period.

Fourier transform for non periodic $f (x)$ is (sum above becomes integral) $\begin{aligned} f (x) & = \frac{1}{2 π} \int_{- \infty}^{\infty} c_{k} e^{i k x} d k \\ c_{k} & = \int_{- \infty}^{\infty} f (x) e^{- i k x} d x \end{aligned}$

This gives rise to $δ (x - x^{'}) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{i k (x - x^{'})} d k$

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