On the asymptotic values of meromorphic functions in the $n$-fold punctured plane

Arturo Fernández Arias

doi:10.3842/umzh.v76i11.7882

Arturo Fernández Arias Dpto Matemáticas Fundamentales, Facultad de Ciencias, UNED, Madrid, Spain

DOI: https://doi.org/10.3842/umzh.v76i11.7882

Keywords: Asymptotic path, Asymptotic value, critical value

Abstract

UDC 517.5

We present some results on the asymptotic paths and asymptotic values for meromorphic functions in the $n$-fold extended punctured plane $\widehat{\mathbb{C}}\backslash\{p_{1},\ldots,p_{n}\} .$ These results extend some classical results obtained for analytic and meromorphic functions in the complex plane $\mathbb{C}.$ In particular, in this more general setting, we give a version of F. Iversen's result on the existence of asymptotic values for entire functions in the plane. We also obtain a bound for the number of isolated directly critical singularities of a meromorphic function of finite order $k$ and a finite number of poles.

References

L. V. Ahlfors, Über die asymptotische Werte der meromorphen Funktionen endlicher Ordnung, Acta Acad. Aboensis Math. Phys., 6, № 9 (1932).

A. Denjoy, Sur les fonctions entières de genre fini, Comptes Rend. Acad. Sci. Paris, 145 (1907).

A. Kondratyuk, I. Laine, Meromorphic functions in circular domains, Methods in Complex Analysis, Report Series, № 10, University of Joensuu (2005).

F. Iversen, Recherches sur les fonctions inverses des fonctions méromorphes, Thèse, Helsingfors (1914).

B. Ja. Levin, Distribution of zeros of entire functions, Transl. Math. Monogr., 5 (1980).

R. Nevanlinna, Analytic functions, Springer-Verlag (1970). DOI: https://doi.org/10.1007/978-3-642-85590-0