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)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-2014-12370-7
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., An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. MR 510197 (80c:30001)
  • [2] Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954. MR 0068627 (16,914f)
  • [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 (35 #730)
  • [6] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Univ. Press., 4th edition, 1927. Reprinted 1962. MR 0178117 (31 #2375)
  • [7] A. H. Zemanian, Distribution theory and transform analysis, An introduction to generalized functions, with applications, 2nd ed., Dover Publications Inc., New York, 1987. MR 918977 (88h:46081)

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
Email: vicente.munoz@mat.ucm.es

Ricardo Pérez Marco
Affiliation: CNRS, LAGA UMR 7539, Université Paris XIII, 99, Avenue J.-B. Clément, 93430-Villetaneuse, France
Email: ricardo@math.univ-paris13.fr

DOI: https://doi.org/10.1090/S0002-9939-2014-12370-7
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

American Mathematical Society