On Shimoda's Theorem

A. A. Atamuratov ; K. K. Rasulov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 33 (2023) no. 1, pp. 17-31

Résumé

The present work is devoted to Shimoda's Theorem on the holomorphicity of a function $f(z,w)$ which is holomorphic by $w\in V$ for each fixed $z\in U$ and is holomorphic by $z\in U$ for each fixed $w\in E$, where $E\subset V$ is a countable set with at least one limit point in $V$. Shimoda proves that in this case $f(z,w)$ is holomorphic in $U\times V$ except for a nowhere dense closed subset of $U\times V$. We prove the converse of this result, that is for an arbitrary given nowhere dense closed subset of $U$, $S\subset U$, there exists a holomorphic function, satisfying Shimoda's Theorem on $U\times V\subset {\mathbb C}^{2}$, that is not holomorphic on $S\times V$. Moreover, we observe conditions which imply empty exception sets on Shimoda's Theorem and prove generalizations of Shimoda's Theorem.

Détail

Keywords: Hartogs's phenomena, Shimoda's Theorem, separately holomorphic functions, power series.

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