Residue of 1-form


Something like Picard for 1-forms ★★

Author(s): Elsner

\begin{conjecture} Let $D$ be the open unit disk in the complex plane and let $U_1,\dots,U_n$ be open sets such that $\bigcup_{j=1}^nU_j=D\setminus\{0\}$. Suppose there are injective holomorphic functions $f_j : U_j \to \mathbb{C},$ $j=1,\ldots,n,$ such that for the differentials we have ${\rm d}f_j={\rm d}f_k$ on any intersection $U_j\cap U_k$. Then those differentials glue together to a meromorphic 1-form on $D$. \end{conjecture}

Keywords: Essential singularity; Holomorphic functions; Picard's theorem; Residue of 1-form; Riemann surfaces

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