- Show that if each \(f_n\) is holomorphic on \(\Omega\) and \(F \definedas \sum f_n\) converges uniformly on every compact subset of \(\Omega\), then \(F\) is holomorphic.
Solution
Goal: If each \(f_n\) is holomorphic on \(\Omega\) and \(F \definedas \sum_n f_n\) converges uniformly on every compact subset of \(\Omega\), show \(F\) is holomorphic on \(\Omega\).
1. \(F\) is continuous on \(\Omega\).
Uniform convergence on compact subsets: fix \(z_0 \in \Omega\) and a compact neighborhood \(K\) of \(z_0\) inside \(\Omega\); the partial sums \(S_N = \sum_{n \leq N} f_n\) are continuous, and \(S_N \rightrightarrows F\) on \(K\), so \(F\) is continuous on \(K\), hence at \(z_0\).
2. For every closed triangle \(\Delta \subseteq \Omega\), \(\int_\Delta F = 0\).
Since \(S_N \rightrightarrows F\) on the compact set \(\Delta\), \(\int_\Delta F = \lim_N \int_\Delta S_N = \lim_N \sum_{n\leq N}\int_\Delta f_n = \lim_N \sum_{n \leq N} 0 = 0\); each \(\int_\Delta f_n = 0\) by the Cauchy–Goursat theorem applied to the holomorphic \(f_n\).
3. \(F\) is holomorphic on \(\Omega\).
By Morera’s theorem, a continuous function on an open set with vanishing integrals over all closed triangles is holomorphic; 1 gives continuity and 2 gives the vanishing integrals.
4. Q.E.D.
3 is the claim.