antiderivative of complex function

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

$$\frac{d}{dz}F(z)=f(z).$$

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

  $F(z):={\displaystyle {\int }_{{\gamma }_z}}f(t)𝑑t$ (1)

  where the path ${\gamma }_z$ begins at a fixed point $z_0$ of $D$ and ends at the point $z$ of $D$, is independent of the path ${\gamma }_z$ for each value of $z$, then (1) defines an analytic function $F$ with domain $D$.  This function is an antiderivative of $f$ in $D$, i.e. (http://planetmath.org/Ie) at all points of $D$, the condition

  $$\frac{d}{dz}{\int }_{{\gamma }_z}f(t)𝑑t=f(z)$$

  is true.

• If $f$ is an analytic function in a simply connected open domain $U$, then $f$ has an antiderivative in $U$, e.g. (http://planetmath.org/Eg) the function $F$ defined by (1) where the path ${\gamma }_z$ is within $U$.  If $\gamma$ lies within $U$ and connects the points $z_0$ and $z_1$, then

  $${\int }_{\gamma }f(z)𝑑z=F(z_1)-F(z_0),$$

  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