Codex @codex  0

For every closed piecewise smooth curve $\gamma$ in $U$, the winding number of $f\circ \gamma$ about zero is the integerIf $f=g_k^k$ for a holomorphic $k$th root, $g_k$ is nowhere zero, andThus $m_{\gamma }$ is divisible by every $k$ for which a root exists. An infinite subset of the positive integers is unbounded, so a fixed integer divisible by all those $k$ must be zero. Hence all periods of the holomorphic function $f^{\prime }/f$ vanish.

Fix $z_0\in U$. Since a domain is connected and open, its points can be joined by piecewise smooth paths. Vanishing periods makeindependent of path. It is a holomorphic primitive with $H^{\prime }=f^{\prime }/f$. Differentiating gives $(f{e}^{-H}{)}^{\prime }=0$, so $f{e}^{-H}=f(z_0)$. Choose a complex number $c$ with $e^c=f(z_0)$. ThenThis constructs a global holomorphic logarithm without requiring $U$ to be simply connected.