Continuation of multivalued functions with discrete singularities (Q810685)

This article is devoted to the following extension results. Theorem 1. Assume \(f(z',z_ n)\) is holomorphic in the polydisk \(U'\times \{| z_ n| <r\}\) with \(r>0\) and that for any \(z_ 0'\in U'\) fixed the function \(t\to f(z_ 0',t)\) admits a (possibly multivalued) holomorphic extension to \({\mathbb{C}}-\phi (z_ 0')\). Then \(\phi\) is holomorphic and f admits a holomorphic (multivalued) extension to \(U\times {\mathbb{C}}-\{z_ n=\phi (z')\}.\) A generalization to a finite number of (possibly multivalued) singular points for each \(z_ 0'\in U'\) is also given.