---
tag: 02AC7JR6G6
kind: statement
role: theorem
name: Holomorphic functions on a disc have primitives
aliases: [existence-of-primitives-on-a-disc]
msc: ["30E20"]
uses: [0DG0Z72CQD, 0BRG85WXYC, 00FWNP9TXY]
sources:
  - url: https://en.wikipedia.org/wiki/Antiderivative_(complex_analysis)
    locator: Existence
  - url: https://en.wikipedia.org/wiki/Antiderivative_(complex_analysis)
    locator: Uniqueness
---

## Statement

Let $c \in \mathbb{C}$, let $R \in (0, \infty]$, and let $D = \{z : |z - c| < R\}$ be the open disc with center $c$ and radius $R$.
Let $f : D \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.
Then there is a holomorphic function $F : D \to \mathbb{C}$ with $F'(z) = f(z)$ for each $z \in D$.

## Remarks

One primitive is $F(z) = \int_{[c, z]} f(w) \, dw$, the [[contour integral|00FWNP9TXY]] along the segment from $c$ to $z$. [[Goursat's theorem|0BRG85WXYC]] for triangles shows that $F' = f$.
Two primitives of $f$ on $D$ differ by a constant, because $D$ is connected.
The disc can be replaced by a convex open set, or more generally by a simply connected open set.
The result fails on $\mathbb{C} \setminus \{0\}$: the function $1/z$ has no primitive there, by the [[integral of 1/z|0SXZ68HV6F]].

## Explanations

### L1 human

On a disc, each holomorphic function is the derivative of another holomorphic function.
You can build it by integrating along straight lines from the center, as in the real case $F(x) = \int_c^x f(t) \, dt$.