Definition:Holomorphic Function/Complex Plane

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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 $U$.

Also defined as

A holomorphic function is sometimes defined as continuously differentiable on the open set $U$.

By Holomorphic Function is Continuously Differentiable, the two definitions 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.

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

Also see

  • Results about holomorphic functions can be found here.