Definition:Holomorphic Function

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Complex Function

Let $U\subset\C$ 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 $U$.

Vector-Valued Function

Also defined as

A holomorphic function is sometimes defined as continuously differentiable. By Goursat's Theorem, the two are equivalent.

Also known as

Some authors refer to a holomorphic function on an open set of $\C$ as a (complex) analytic function. This is because, by Holomorphic Function is Analytic, they are equivalent.

Also see