I am attempting to solve the problem listed here: Laurent Series on a square annulus
and I tried the following approach because I didn't quite understand the accepted answer. I am wondering if this is valid or if I've gone off the rails somehow.
Proof:
$S_7$ is simply connected and not all of $\mathbb C$ so by the Riemann mapping theorem, there exists $\phi:S_7\to\mathbb D$ a conformal map where $\phi(0)=0$ and $\mathbb D$ is the open unit disk. (Sketchy) Since $\phi$ takes the boundary of $S_7$ to a circle and since $\phi$ is conformal, $\phi$ takes the boundary of $S_6$ to a circle too. That is, $\phi(S_6)=\mathbb D_r$ where $r<1.$ We can now see that $\phi\circ f$ is analytic on the annulus $A_{r,1}$ and by Laurent's theorem there exists a unique decomposition into series: $\phi\circ f(z)=\sum a_nz^n +\sum b_nz^n$ where the first part is analytic on $\mathbb D$ and the second is analytic on $\mathbb C\setminus \overline{\mathbb{ D_r}}.$
Then, we could set $f_+=\phi^{-1}(\sum a_nz^n)$ and $f_-=\phi^{-1}(\sum b_nz^n)$?
Question 1: Is this proof valid?
Question 2: I don't see how the answer from the cited question can solve part b) of the problem: "How many different ways are there of writing $f$ as the sum of $f_+$ and $f_−$ as in part (a)? Find all possible ways." It makes more sense to me to try and cite uniqueness in Laurent's Theorem.