Derivative of Sequence of Holomorphic Functions
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Theorem
Let $U$ be an open, connected subset of $\C$.
Let $\left \langle{f_n}\right \rangle$ be a sequence of holomorphic mappings $f_n: U \to \C$.
Let $\left \langle{f_n}\right \rangle$ converge pointwise to some function $f: U \to \C$.
Let $\left \langle{f_n}\right \rangle$ converge uniformly on compact subsets of $U$.
Then $f$ is holomorphic on $U$.
Further, the sequence of derivatives $\left \langle{f_n'}\right \rangle$ converges to $f'$ on $U$.
This convergence is uniform on compact subsets of $U$.
Proof
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Sources
- 2005: Eberhard Freitag and Rolf Busam: Complex Analysis: $3$