Convergence of power series and coefficients

Question

I am studying complex analysis out of Gamelin, and I'm stuck on this problem in the text.

Let $f_m(z)$ be a sequence of analytic functions converging uniformly to some $f(z)$ over $|z| < R$. Because they are analytic, I write the $f_m(z)$ as series $\sum_{k} a_{m,k}z^k$. By the uniform convergence, $f$ is analytic over domain and has a series representation $\sum_{k} a_k z^k$. Question: Prove that for each $k$ that $a_{k,m} \rightarrow a_{k}$.

My first idea for starting this was to imitate the proof of the uniform limit theorem (i.e. uniform limit of continuous functions is continuous), but dealing with series made that seem to complicated. I appreciate any hint/help for this one.

Answer 1

Since you have that $f_m$ converges to $f$ uniformly and have already shown $f$ to be analytic, note that

\begin{eqnarray} \lim a_{k,n} &&= \lim \frac{1}{2\pi i}\int_{C} \frac{f_n(\zeta)}{(\zeta - z)^{k+1}}d\zeta\\ &&= \frac{1}{2\pi i} \int_C \lim \frac{f_n(\zeta)}{(\zeta -z)^{k+1}}d\zeta\\ &&= \frac{1}{2\pi i} \int_C \frac{f(\zeta)}{(\zeta -z)^{k+1}}d\zeta \\ &&= a_k \end{eqnarray}

where $z$ is the point you are basing you power series at and $C$ is some circle in your domain around $z$.