antiderivative of complex function

By the of a complex function f in a domain D of , we every complex function F which in D satisfies the condition

  • If f is a continuousMathworldPlanetmathPlanetmath complex function in a domain D and if the integral

    F(z):=γzf(t)𝑑t (1)

    where the path γz begins at a fixed pointPlanetmathPlanetmath z0 of D and ends at the point z of D, is independent of the path γz for each value of z, then (1) defines an analytic functionMathworldPlanetmath F with domain D.  This function is an antiderivative of f in D, i.e. ( at all points of D, the condition


    is true.

  • If f is an analytic function in a simply connected open domain U, then f has an antiderivative in U, e.g. ( the function F defined by (1) where the path γz is within U.  If γ lies within U and connects the points z0 and z1, then


    where F is an arbitrary antiderivative of f in U.

Title antiderivative of complex function
Canonical name AntiderivativeOfComplexFunction
Date of creation 2014-02-23 15:09:20
Last modified on 2014-02-23 15:09:20
Owner Wkbj79 (1863)
Last modified by pahio (2872)
Numerical id 10
Author Wkbj79 (2872)
Entry type Definition
Classification msc 30A99
Classification msc 03E20
Synonym complex antiderivative
Related topic Antiderivative
Related topic CalculationOfContourIntegral