Definition:Holomorphic Function

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Definition

Complex Function

Let $U \subseteq \C$ be an open set.

Let $f : U \to \C$ be a complex function.

Then $f$ is holomorphic in $U$ if and only if $f$ is differentiable at each point of $U$.

We also say that $f$ is complex-differentiable in $U$.

Vector-Valued Function

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Also defined as

A holomorphic function is sometimes defined as continuously differentiable.

By Holomorphic Function is Continuously Differentiable, the two are equivalent.

Also known as

Some authors refer to a holomorphic function on an open set of $\C$ as an analytic (complex) function.

This is because, by Holomorphic Function is Analytic, they are equivalent.

This theorem requires a proof. In particular: holomorphic iff analytic 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.

Sometimes the term regular function can be seen, which means the same thing.

Also see

• Definition:Meromorphic Function
• Definition:Analytic Complex Function

• Results about holomorphic functions can be found here.