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By the antiderivative of a complex function $f$ in a domain $D$ of $\mathbb{C}$, we mean 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$\displaystyle F(z):=\int_{{\gamma_{z}}}f(t)\,dt$ (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. at all points of $D$, the condition$\frac{d}{dz}\int_{{\gamma_{z}}}f(t)\,dt=f(z)$ 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. 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 $U$.

## Mathematics Subject Classification

30A99*no label found*03E20

*no label found*

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