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: $3$