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.
Sometimes the term regular function can be seen, which means the same thing.
Also see
- Results about holomorphic functions can be found here.