This question involves the following homework problem:
PROBLEM
Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form
$$f(z)=\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$$
with
$$c_n= \int_0^1 f(x+iy) e^{-2\pi i n (x+iy)}dx$$
for any value $y>0$.
HINT
Show that there is an analytic function $f^*$ on a disk from which the origin is deleted such that
$$f^*(e^{2\pi i z })=f(z)$$
what is the Laurent series for $f^*$? Abbreviate $q=e^{2\pi i z }$.
MY TRY
I thought that the extension to the real line should also be analytic:
$$g(x):= \lim_{y\text{ to }0}f(x+iy)$$
and then by periodicity and the fact that $f,g$ are analytic this function is equal to its Fourier series everywhere (with possibly complex coefficients). Then by again the fact that $f,g,\sin$ and cos are analytic this is then equal to the same series using the complex series definition for $\sin$ and $\cos$. I have the idea that I am still missing something, since the hint implies a totally different approach. Thanks!