Show that if each \(f_n\) is holomorphic on an open set \(\Omega \subseteq \mathbb{C}\) and \(F \coloneqq \sum_{n=1}^\infty f_n\) converges uniformly on every compact subset of \(\Omega\), then \(F\) is holomorphic on \(\Omega\).
Solution
Let \[ S_N=\sum_{n=1}^N f_n. \] Each \(S_N\) is holomorphic on \(\Omega\).
1. The limit \(F\) is continuous. For every \(z_0\in\Omega\), choose \(r>0\) with\[ \overline{D_r(z_0)}\subset\Omega. \]The convergence \(S_N\to F\) is uniform on this compact disk, so \(F\) is a uniform limit of continuous functions there and is therefore continuous near \(z_0\).
2. Let \(T\) be any closed triangle contained in \(\Omega\). Since each \(S_N\) is holomorphic on a neighborhood of \(T\), Cauchy’s theorem gives\[ \int_{\partial T}S_N(z)\,dz=0. \]The boundary \(\partial T\) is compact, so \(S_N\to F\) uniformly on \(\partial T\). Hence\[ \int_{\partial T}F(z)\,dz =\lim_{N\to\infty}\int_{\partial T}S_N(z)\,dz =0. \]
3. Morera’s theorem now implies that \(F\) is holomorphic on \(\Omega\).
4. Moreover, once holomorphy is known, Cauchy’s integral formula on nested compact disks shows that for every \(k\ge1\),\[ F^{(k)}=\sum_{n=1}^\infty f_n^{(k)} \]with locally uniform convergence of the derivative series.