Antiderivative (complex analysis)

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 <math>U</math> in the complex plane and a function <math>g:Uto mathbb C,</math> the antiderivative of <math>g</math> is a function <math>f:Uto mathbb C</math> that satisfies <math>frac=g</math>.

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 <math>U</math> 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 <math>U</math> (those constants need not be equal).

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

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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