Holomorphic function

Holomorphic function is concept of the theory of functions of complex variables that refers the the existence of the derivative.

Definition

Assume, for any $z \in C\subseteq \mathbb C$, there is defined function $f(z) \in \mathbb C$ such that for any $z \in C$ there exist the derivative

$\displaystyle f'(z)= \lim_{t \rightarrow 0,~ t\in \mathbb C}~ \frac{f(z\!+\!t)-f(z)}{t}

$

Then, function $f$ is called holomorphic on $C$.

References

http://en.citizendium.org/wiki/Holomorphic_function

http://en.wikipedia.org/wiki/Holomorphic_function

http://www.proofwiki.org/wiki/Definition:Holomorphic_Function

http://www.proofwiki.org/wiki/Equivalence_of_Definitions_for_Analytic_Function