[Math] Can a function be analytic and independent of path on a domain $D$, but have no anti-derivative on $D$

Question

My class notes seem to have a contradiction. I want to know if there's an error in my notes or if I am misunderstanding.

Firstly there is a theorem that if a function $f$ is analytic on a (edit: simply-connected) domain $D$, then a line integral of $f$ in $D$ is independent of path.

Secondly there is a claim that a function can be analytic on a domain, but has no anti-derivative on that domain.

Finally I read that a function has an anti-derivative on a domain if and only if the integral of the function along a contour is path independent in D.

So it seems like I can have a function $f$, which is analytic on a domain $D$ but contains no anti-derivative on $D$. Then since $f$ is analytic, its integral must be path independent on $D$. Then $f$ has an anti-derivative on $D$ which is a contradiction.

Answer 1

The first of your theorems is incorrect (or is missing assumptions):

if a function $f$ is analytic on a domain $D$, then a line integral of $f$ in $D$ is independent of path.

is not true for every domain $D$. It is true if you assume that $D$ is simply connected.