On Weierstraß’s theorem of the meromorphic functions with non- isolated essentially singular points. (Chinese. English summary) Zbl 0591.30021
From the author’s summary:
Theorem. Let E be a compact subset of the complex plane and let f(z) be a meromorphic function outside E such that the essentially singular points of f(z) are non-void. Then a necessary and sufficient condition that there exists a sequence \(\{z_ n\}^{\infty}_{n=1}\subset S^ 2/E\) for an arbitrary complex number A such that \(\lim_{d(z_ n,E)\to 0}f(z_ n)=A\) is that the analytic capacity of E vanishes, where \(d(z_ n,E)\) is the distance of \(z_ n\) with E.
Remark: The title of the paper mentions non-isolated essentially singular points of a meromorphic function, while in the abstract and the proof of the main theorem an essentially singular point is dealt with.
Theorem. Let E be a compact subset of the complex plane and let f(z) be a meromorphic function outside E such that the essentially singular points of f(z) are non-void. Then a necessary and sufficient condition that there exists a sequence \(\{z_ n\}^{\infty}_{n=1}\subset S^ 2/E\) for an arbitrary complex number A such that \(\lim_{d(z_ n,E)\to 0}f(z_ n)=A\) is that the analytic capacity of E vanishes, where \(d(z_ n,E)\) is the distance of \(z_ n\) with E.
Remark: The title of the paper mentions non-isolated essentially singular points of a meromorphic function, while in the abstract and the proof of the main theorem an essentially singular point is dealt with.
Reviewer: C.-C.Yang
MSC:
| 30C85 | Capacity and harmonic measure in the complex plane |
