Laurent series VS Fourier series.

Question

Is there a link between Fourier and Laurent series ? Let for example $$f(z)=\sum_{n\in\mathbb Z}a_nz^n.$$ We have that the $a_n$ are given by $a_n:=\frac{f^{(n)}(0)}{n!}=\frac{1}{2i\pi }\int_\gamma \frac{f(z)}{z^{n+1}}dz,$ where $\gamma $ is a close curved s.t. the inside contain $0$. In particular, if I set $z=e^{i\theta }$, then $$g(\theta )=f(e^{i\theta })=\sum_{k\in\mathbb Z}a_ne^{in\theta },$$ and thus look to be the Fourier series of a function $g$, and thus $a_n=\frac{1}{2\pi}\int_0^{2\pi}g(\theta )e^{-in\theta }d\theta .$ I'm a little bit confuse... is there a strong link between those two series ?

Answer 1

See Laurent Series Yield Fourier Series and From Laurent to Fourier.