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 \(C\).