Derivative of Sequence of Holomorphic Functions

Theorem

Let $U$ be an open, connected subset of $\C$.

Let $\sequence {f_n}$ be a sequence of holomorphic functions $f_n: U \to \C$.

Let $\sequence {f_n}$ converge pointwise to some function $f: U \to \C$.

This definition needs to be completed. In particular: The definition of converge pointwise above is in the real domain only. We need a version for complex analysis. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{DefinitionWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Let $\sequence {f_n}$ converge uniformly on compact subsets of $U$.

Then $f$ is holomorphic on $U$.

Further, the sequence of derivatives $\sequence {f_n'}$ converges to $f'$ on $U$.

This convergence is uniform on compact subsets of $U$.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2005: Eberhard Freitag and Rolf Busam: Complex Analysis: $3$