# antiderivative of complex function

By the of a complex function $f$ in a domain $D$ of $\mathbb{C}$, 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

 $\displaystyle F(z):=\int_{\gamma_{z}}f(t)\,dt$ (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 AntiderivativeOfComplexFunction 2014-02-23 15:09:20 2014-02-23 15:09:20 Wkbj79 (1863) pahio (2872) 10 Wkbj79 (2872) Definition msc 30A99 msc 03E20 complex antiderivative Antiderivative CalculationOfContourIntegral