Let $G\subset C$ be open and simply connected, and $A\subset G$ a discrete subset of $G$. Prove that a holomorphic function $f$ on $G/A$ has an antiderivative on $G/A$ if and only if $\operatorname{res}_a(f)=0$ for all $a\in A$.

For $(\Leftarrow)$, from Residue theorem,we know that for any simple closed curve $\gamma$ we have $\int_\gamma f(z) \, dz=0,$ then we can define $F(z)=\int^z_{z_0}f(z) \, dz$ for any fixed point $z_0$ in $G/A$. And this is the antiderivative of $f(z)$.

And I don't know how to approach the $(\Rightarrow)$.

  • $\begingroup$ You want $G\setminus A.$ $\endgroup$ – zhw. Aug 25 '16 at 23:57

You did the difficult direction.

If $F$ is an antiderivative of $f$, then $\int_\gamma f(z)\,dz = 0$ along any closed curve in $G\setminus A$ by the complex version of the fundamental theorem of calculus. (And the residue at $a$ is defined as such an integral.)


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