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
As such, this concept is the complex-variable version of the antiderivative of a real-valued function.
One can characterize the existence of antiderivatives via path integrals in the complex plane, much like the case of functions of a real variable.
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.
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.
Various versions of Cauchy integral theorem, an underpinning result of Cauchy function theory, which makes heavy use of path integrals, gives sufficient conditions under which, for a holomorphic g, vanishes for any closed path γ (which may be, for instance, that the domain of g be simply connected or star-convex).
First we show that if f is an antiderivative of g on U, then g has the path integral property given above.
Next we show that if g is holomorphic, and the integral of g over any path depends only on the endpoints, then g has an antiderivative.
With this assumption, fix a point z0 in U and for any z in U define the function where γ is any path joining z0 to z.
Such a path exists since U is assumed to be an open connected set.
The function f is well-defined because the integral depends only on the endpoints of γ.
Then for every w other than z within this disk where [z, w] denotes the line segment between z and w. By continuity of g, the final expression goes to zero as w approaches z.