| Peer-Reviewed

A Logarithmic Derivative of Theta Function and Implication

Received: 26 June 2015     Accepted: 28 June 2015     Published: 30 June 2016
Views:       Downloads:
Abstract

In this paper we establish an identity involving logarithmic derivative of theta function by the theory of elliptic functions. Using these identities we introduce Ramanujan’s modular identities, and also re-derive the product identity, and many other new interesting identities.

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 5-1)

This article belongs to the Special Issue Mathematical Aspects of Engineering Disciplines

DOI 10.11648/j.pamj.s.2015040501.21
Page(s) 55-59
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Theta Function, Elliptic Function, Logarithmic Derivative

References
[1] W. N .Bailey, A further note on two of Ramanujan’s formulae, Q. J. Math.(Oxford) 3 (1952), pp.158-160.
[2] R. Bellman, A brief introduction to the theta functions, Holt Rinehart and Winston, New York(1961).
[3] B. C. Berndt, Ramanujan’s Notebooks III, Springer-Verlag, New York (1991).
[4] J.M. Borwein and P. B. Borwein, Pi and the AGM- A Study in Analytic Number Theory and Computational Complexity, Wiley, N.Y., 1987.
[5] J.M. Borwein, P. B. Borwein and F. G. Garvan, Some cubic modular Indentities of Ramanujan, Trans. of the Amer. Math. Soci., Vol. 343, No. 1 (May, 1994), pp.35-47
[6] J. A. Ewell, On the enumerator for sums of three squares, Fibon.Quart.24(1986), pp.151-153.
[7] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. Cambridge Univ. Press, 1966
[8] Li-Chien Shen, On the Additive Formulae of the Theta Functions and a Collection of Lambert Series Pertaining to the Modular Equations of Degree 5, Trans. of the Amer. Math. Soci. Vol. 345, No. 1 (Sep., 1994), pp.323-345.
Cite This Article
  • APA Style

    Yaling Men, Jiaolian Zhao. (2016). A Logarithmic Derivative of Theta Function and Implication. Pure and Applied Mathematics Journal, 4(5-1), 55-59. https://doi.org/10.11648/j.pamj.s.2015040501.21

    Copy | Download

    ACS Style

    Yaling Men; Jiaolian Zhao. A Logarithmic Derivative of Theta Function and Implication. Pure Appl. Math. J. 2016, 4(5-1), 55-59. doi: 10.11648/j.pamj.s.2015040501.21

    Copy | Download

    AMA Style

    Yaling Men, Jiaolian Zhao. A Logarithmic Derivative of Theta Function and Implication. Pure Appl Math J. 2016;4(5-1):55-59. doi: 10.11648/j.pamj.s.2015040501.21

    Copy | Download

  • @article{10.11648/j.pamj.s.2015040501.21,
      author = {Yaling Men and Jiaolian Zhao},
      title = {A Logarithmic Derivative of Theta Function and Implication},
      journal = {Pure and Applied Mathematics Journal},
      volume = {4},
      number = {5-1},
      pages = {55-59},
      doi = {10.11648/j.pamj.s.2015040501.21},
      url = {https://doi.org/10.11648/j.pamj.s.2015040501.21},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040501.21},
      abstract = {In this paper we establish an identity involving logarithmic derivative of theta function by the theory of elliptic functions. Using these identities we introduce Ramanujan’s modular identities, and also re-derive the product identity, and many other new interesting identities.},
     year = {2016}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - A Logarithmic Derivative of Theta Function and Implication
    AU  - Yaling Men
    AU  - Jiaolian Zhao
    Y1  - 2016/06/30
    PY  - 2016
    N1  - https://doi.org/10.11648/j.pamj.s.2015040501.21
    DO  - 10.11648/j.pamj.s.2015040501.21
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
    SP  - 55
    EP  - 59
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.s.2015040501.21
    AB  - In this paper we establish an identity involving logarithmic derivative of theta function by the theory of elliptic functions. Using these identities we introduce Ramanujan’s modular identities, and also re-derive the product identity, and many other new interesting identities.
    VL  - 4
    IS  - 5-1
    ER  - 

    Copy | Download

Author Information
  • School of Mathematics, Xianyang Vocational and Technical College, Xianyang, P. R. China

  • School of Mathematics and Informatics, Weinan Teacher`s University, Weinan, P. R. China

  • Sections