I would appreciate if anyone can help me with this question.
My attempt: Let $f(z)=\sum_{m=-\infty}^\infty a_m(z-c)^m$. I divided both sides by $(z-c)^{n+1}$ to get $$\frac{f(z)}{(z-c)^{n+1}}=\sum_{m=-\infty}^\infty a_m(z-c)^{m-n-1}.$$ But, since $\gamma$ is a closed curve in $A$ we have for $m\neq n$, $$\int_\gamma a_m(z-c)^{m-n-1}dz=0.$$
This gives $$\int_\gamma\frac{f(z)}{(z-c)^{n+1}}=0$$ and so I am stuck.
