I am trying to prove the following problem: [Source, page 2 problem 3]
Let $S_6$ and $S_7$ be the open squares centered at the origin of side length 6 and 7, respectively. Let $\Gamma_6$ and $\Gamma_7$ be the boundary of $S_6$ and $S_7$, respectively, and let $\Omega$ be the region between $\Gamma_6$ and $\Gamma_7$.
Let $f$ be a function that is holomorphic in an open neighborhood of $\overline \Omega$. Prove that there are functions $f_+$ and $f_-$ where $f_+$ is holomorphic in $S_7$, $f_-$ is holomorphic in $\mathbb{C} \setminus \overline{S_6}$, and $$f(z) = f_+(z) + f_-(z),$$ for all $z \in \Omega$.
My goal is to write $f(z)$ as $\sum_{n=0}^\infty a_n z^n + \sum_{n=1}^{\infty} \frac{b_n}{z^n}$, where the power series $\sum_{n=0}^\infty a_n z^n$ converges in a radius of $\frac{7}{2}\sqrt{2}$ and the Laurent series $\sum_{n=1}^{\infty} \frac{b_n}{z^n}$ converges outside a radius of 3. I begin with what I think is obvious $$a_n = \frac{1}{2\pi i}\int_{\Gamma_{6.5}} \frac{f(\zeta)}{\zeta^{n+1}} d\zeta.$$ $$b_n = \frac{1}{2\pi i}\int_{\Gamma_{6.5}} \frac{f(\zeta)}{\zeta^{-n+1}} d\zeta.$$ (Here $\Gamma_{6.5}$ is the boundary of the square with side length 6.5).
Let $f_+(z) = \sum_{n=0}^\infty a_n z^n$, and $f_-(z) = \sum_{n=1}^{\infty} \frac{b_n}{z^n}$. I suspect that these are the functions I need, but I'm struggling to show that $f_+$ and $f_-$ are defined on $\Omega$ (i.e., the radii of convergence of $f_+(z)$ and $f_-(\frac{1}{z})$ are at least $\frac{7}{2}\sqrt{2}$ and at most 3 respectively) and that $f = f_- + f_+$.
Any hints or solutions are appreciated!
The first answer is you are being too ambitious in what you want to show. If you could do that, it would show any holomorphic $f$ on $\Omega$ extends holomorphically to an annular region strictly larger than $\Omega.$ That's not going to work.
Let $\Gamma_r$ be the square of side $r$ centered at the origin. Suppose $z \in \Omega.$ Then for $6<r_1<r_2<7$ with $r_1$ close to $6$ and $r_2$ close to $7,$ we have
$$f(z) = \frac{1}{2\pi i}\int_{\Gamma_{r_2}}\frac{f(\zeta)}{\zeta -z}\, d\zeta - \frac{1}{2\pi i}\int_{\Gamma_{r_1}}\frac{f(\zeta)}{\zeta -z}\, d\zeta.$$
The first integral defines a holomorphic function in the interior of $\Gamma_{r_2},$ the second integral defines a holomorphic function in the exterior of $\Gamma_{r_1}.$ Play around with that idea for a bit.