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Holomorphic functions

If is an open subset of   and is a complex valued function on and , then is said to be complex differentiable at a point if the limit of the difference quotient

(1)

taken over all sequences of complex numbers approaching exists and is finite.  If this is the case the limit is denoted by , or ,  or . The complex  differentiability shares many properties with real differentiability as for instance,  it is linear and obeys the known product, quotient and chain rules. However, it is much stronger than in the real domain analysis .  One of the most significant differences is complex differentiability implies that the function is infinitely often complex differentiable and can be described by its Taylor series. This in other words gives one of the most important theorems of complex analysis saying that a holomorphic functions is analytic.

If is complex differentiable at every point , we say that f is holomorphic on . For instance,  every polynomial in with complex coefficients is holomorphic on the whole complex plane .  Another example is the principal branch of the logarithm function which is holomorphic on the set .  An example of a function which is not holomorphic is the complex conjugation .

The name holomorphic is derived from the Greek holos meaning "whole" and morphe meaning "form".

We say that is holomorphic at the point   if it is holomorphic on some neighborhood of this point.

We ay that is holomorphic in an open set if it is holomorphic at every point of . We say that is holomorphic on some non-open set if it can be prolonged to open set containing and it is holomorphic in .

We say that is holomorphic at infinity, if the function is holomorphic at .

A function that is holomorphic on the whole complex plane is called an entire function. If an entire function has a singularity or an essential singularity at the complex point at infinity then it called a transcendental entire function.

The sum and product of finitely many holomorphic functions is again a holomorphic function.

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