Primitive of an holomorphic function

Question

Why does an holomorphic function have a primitive in a simply connected space? Also, it have a primitive only in a simply connected space?

Answer 1

Let $f(z)$ be your holomorphic function, defined on $U \subset \mathbb C$. Define your primitive by picking a point $z_0 \in \mathbb C$, and writing $$ F(z) = \int_{z_0}^z f(w) dw. $$ But for this to make sense, this integral needs to be independent of the choice of integration path between $z_0$ and $z$!

If $U$ is simply connected, the integral is indeed independent of the path, by Cauchy's theorem.

That's not to say that it's always impossible to define primitives on non-simply-connected spaces. For example, if $U = \mathbb C - \{0\}$, then $f(z) = 1$ has a primitive, namely $F(z) = z$, but $f(z) = 1/z$ is an example of a function which does not.