antiderivative of complex function
By the of a complex function $f$ in a domain $D$ of $\u2102$, we every complex function $F$ which in $D$ satisfies the condition
$$\frac{d}{dz}F(z)=f(z).$$ 

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If $f$ is a continuous^{} complex function in a domain $D$ and if the integral
$F(z):={\displaystyle {\int}_{{\gamma}_{z}}}f(t)\mathit{d}t$ (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
$$\frac{d}{dz}{\int}_{{\gamma}_{z}}f(t)\mathit{d}t=f(z)$$ is true.

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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
$${\int}_{\gamma}f(z)\mathit{d}z=F({z}_{1})F({z}_{0}),$$ where $F$ is an arbitrary antiderivative of $f$ in $U$.
Title  antiderivative of complex function 

Canonical name  AntiderivativeOfComplexFunction 
Date of creation  20140223 15:09:20 
Last modified on  20140223 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 