Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


On the genus of meromorphic functions

Authors: Vicente Muñoz and Ricardo Pérez Marco
Journal: Proc. Amer. Math. Soc. 143 (2015), 341-351
MSC (2010): Primary 30D30; Secondary 30B50, 30D15
Published electronically: September 15, 2014
MathSciNet review: 3272759
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We define the class of Left Located Divisor (LLD) meromorphic functions, their vertical order $ m_0(f)$ and their convergence exponent $ d(f)$. When $ m_0(f)\leq d(f)$ we prove that their Weierstrass genus is minimal. This explains the phenomena that many classical functions have minimal Weierstrass genus, for example, Dirichlet series, the $ \Gamma $-function, and trigonometric functions.

References [Enhancements On Off] (What's this?)

  • [1] Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197
  • [2] Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954. MR 0068627
  • [3] G. H. Hardy and M. Riesz, The general theory of Dirichlet's series, Dover, 1915.
  • [4] V. Muñoz and R. Pérez-Marco, Unified treatment of explicit and trace formulas via Poisson-Newton formula, arXiv:1309.1449, 2013.
  • [5] Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966 (French). MR 0209834
  • [6] E. T. Whittaker and G. N. Watson, A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions, Fourth edition. Reprinted, Cambridge University Press, New York, 1962. MR 0178117
  • [7] A. H. Zemanian, Distribution theory and transform analysis, 2nd ed., Dover Publications, Inc., New York, 1987. An introduction to generalized functions, with applications. MR 918977

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30D30, 30B50, 30D15

Retrieve articles in all journals with MSC (2010): 30D30, 30B50, 30D15

Additional Information

Vicente Muñoz
Affiliation: Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain

Ricardo Pérez Marco
Affiliation: CNRS, LAGA UMR 7539, Université Paris XIII, 99, Avenue J.-B. Clément, 93430-Villetaneuse, France

Keywords: Dirichlet series, Poisson-Newton formula, Hadamard factorization.
Received by editor(s): April 30, 2013
Published electronically: September 15, 2014
Additional Notes: The authors were partially supported through Spanish MICINN grant MTM2010-17389.
Communicated by: Franc Forstneric
Article copyright: © Copyright 2014 American Mathematical Society