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Concept
Antiderivative (complex analysis)
Summary
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 in the complex plane and a function the antiderivative of is a function that satisfies .
As such, this concept is the complex-variable version of the antiderivative of a real-valued function.
The derivative of a constant function is the zero function. Therefore, any constant function is an antiderivative of the zero function. If is a connected set, then the constant functions 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 (those constants need not be equal).
This observation implies that if a function has an antiderivative, then that antiderivative is unique up to addition of a function which is constant on each connected component of .
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 γ path from a to b, the path integral
Equivalently,
for any closed path γ.
However, this formal similarity notwithstanding, possessing a complex-antiderivative is a much more restrictive condition than its real counterpart. While it is possible for a discontinuous real function to have an anti-derivative, anti-derivatives can fail to exist even for holomorphic functions of a complex variable. For example, consider the reciprocal function, g(z) = 1/z which is holomorphic on the punctured plane C{0}. A direct calculation shows that the integral of g along any circle enclosing the origin is non-zero. So g fails the condition cited above. This is similar to the existence of potential functions for conservative vector fields, in that Green's theorem is only able to guarantee path independence when the function in question is defined on a simply connected region, as in the case of the Cauchy integral theorem.
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