# Holomorphic function which is not the derivative of a holomorphic function

I don't know how much knowledge of complex analysis is needed to find an open set $U\subset \mathbb{C}$ and a holomorphic function $f\colon U\to \mathbb{C}$, which is not the derivative of a holomorphic function $U\to \mathbb{C}$.
My knowledge is pretty much limited to knowing what a holomorphic function is (roughly speaking, a function that locally looks like a power series).
I appreciate any sort of information helping me to find the required data.

• @user1952009 what about $1/z$ on $\mathbb{C}\setminus\{0\}$? – Mark May 21 '16 at 10:49
• @Mark : I didn't think to "an open set which is not simply connected" right. the Cauchy integral formula ensures that it doesn't exist when $U$ is simply conntected, by proving that holomorphic on $U$ = analytic on $U$. start with $U$ the open unit disk, this way $f$ is explicitly a power series and has an anti derivative – reuns May 21 '16 at 10:50

## 2 Answers

take $U=\{z\in\mathbb{C} : 0<|z|<1\}$ and $f(z)=\frac{1}{z}$

Note that if a holomorphic function $f$ has an antiderivative (also known as a primitive) $F$, this means that $\oint_\gamma f=F(b)-F(a)=0$ for any closed path $\gamma$ (on the domain $F$ is defined, that is). So, look for functions $f$ such that $\oint_\gamma f\neq 0$.