# Derivative of Sequence of Holomorphic Functions

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## 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*: $3$