Holomorphic function

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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 \(C\).

Cauchi–Riemann

Infinite detivatives

Other notations

Examples

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