I want to solve the following problem
The way I thought about this problem is to consider the following:
Let $f^*(z)$ be holomorphic on the whole plane but the origin. So it has laurent series $$f^*(z)=\sum a_n z^n.$$
Let $$b(z)=e^{2\pi i z}.$$
This function has a range of the upper half plane and moreover, has a period of $1$ Moreover, it is analitic and so
$$f^*(b(z))=\sum c_n {e^{2\pi izn}}=\sum c_n q^n$$
is an analitic function on the upper plane of period of $1$ as ($b(z)=q$). Therfore
$$f(x)=f^*(q)$$
for some $f^*(q)$ as described above. Furthermore,
$$c_n =\frac{1}{2\pi i} \oint_{\gamma _r} \frac{f^*(z)}{z^{n+1}}dz$$
for all $r>0$. We can paremeterize by $z=e^{i\pi(x+iy)}$ with $y$ beeing treated as a constant (dependent on the radius) to get $$c_n=\int_{0}^{1}f^*(e^{2\pi z})e^{-2\pi i zn}dx=\int_{0}^{1}f(x+iy)e^{-2\pi i zn}dx$$
as required.
Issues with this: To me this proof holds as one can always find an analitic function on the complex plane and remove the origin from its domain. However, I checked the answers and the solution that they have seems to be quite different (for the first part)
Questions:
$1$) Does this mean that my solution is wrong?
$2$) Can someone help me understand the first part of their solution?
Thank you in advance.
Regarding the first part of the solution: do you know the definition of the complex logarithm? Are you familiar with the notion of winding number, and the reason for $g_{\gamma}(z)$ differing by integer multiples of $2 \pi i$ for different branches of the logarithm?
In any case, the solution explicitly shows the existence of a holomorphic function which injects the punctured plane into all $\mathbf{C}$. The issue with your solution is that you implicitly assume that if there exists some $f^{*}$ such that $f^{*} \circ e^{2 \pi i z} = f(z)$ for any $f(z)$ which is holomorphic on the upper half plane and has period 1. This function is in fact the complex logarithm, and its existence is nontrivial - the solution shows that it is possible to construct a function, and specifically states the domain on which this construction is valid.
A side note: the range of $e^{2 \pi i z}$ is not the upper half plane, but the puncture complex plane $\mathbf{C} \backslash \{ 0 \}$.