## You are here

Homeantiderivative of complex function

## Primary tabs

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

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

- •
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.

- •
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*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

new question: Lorenz system by David Bankom

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith