- Let P be a point in an open set U in C, and suppose that f is a meromorphic function on U with a pole at P . Prove that there is no holomorphic function \(g : U \setminus \{ P \} \to \mathbb { C }\) such that \(e ^ { g ( z ) } = f ( z )\) for all \(z \in U \setminus \{ P \}\)
Solution
1. The logarithmic derivative has a nonzero integral around the pole.
Let \(m\geq1\) be the pole order. Locally, \[ f(z)=(z-P)^{-m}h(z), \] where \(h\) is holomorphic and \(h(P)\ne0\), by the Laurent expansion [@SS03]. Choose \(r>0\) so that \(h\) is holomorphic and nonzero on a neighborhood of \(\overline{D(P,r)}\subset U\). On the punctured disk, \[ \frac{f'(z)}{f(z)}=-\frac{m}{z-P}+\frac{h'(z)}{h(z)}. \] Since \(h'/h\) is holomorphic on the full disk, Cauchy’s theorem and direct circle integration give \[ \int_{|z-P|=r}\frac{f'(z)}{f(z)}\,dz=-2\pi im\ne0 \] with counterclockwise orientation [@SS03].
2. A holomorphic logarithm would make the same integral zero.
1. The primitive is thesingle-valued function \(g\) itself; no simple-connectednessassumption on \(U\setminus\{P\}\) is used.