Let $\Omega$ be a domain and $f$ is a holomorphic function with no zeros in $\Omega$. Suppose for infinitely many positive integer $k$, $f$ has a holomorphic $k$ th root. Prove that there exists a holomorphic function $F$ such that $ f=e^F$.
My try: I guess the argument principle might be useful here. But i don’t know how to prove it. When domain is simply connected, such result can be proved by letting $F(z)=F(z_0)+\int_{z_0}^z\frac{f’}{f}$.
