 Antiderivative (Complex Analysis)

# Antiderivative (complex analysis)

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Description:
In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function g is a function whose complex derivative is g. More precisely, given an open set [itex]U[/itex] in the complex plane and a function [itex]g:Uto mathbb C,[/itex] the antiderivative of [itex]g[/itex] is a function [itex]f:Uto mathbb C[/itex] that satisfies [itex]frac=g[/itex].

As such, this concept is the complex-variable version of the antiderivative of a real-valued function.

## Uniqueness

The derivative of a constant function is zero. Therefore, any constant is an antiderivative of the zero function. If [itex]U[/itex] is a connected set, then the constants are the only antiderivatives of the zero function. Otherwise, a function is an antiderivative of the zero function if and only if it is constant on each connected component of [itex]U[/itex] (those constants need not be equal).

This observation implies that if a function [itex]g:Uto mathbb C[/itex] has an antiderivative, then that antiderivative is unique up to addition of a function which is constant on each connected component of [itex]U[/itex].

## Existence

One can characterize the existence of antiderivatives via path integrals in the complex plane, much like the case of functions of a real variable. Perhaps not surprisingly, g has an antiderivative f if and only if, for every &gamma; path from a...

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