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 |
Theta Function, Elliptic Function, Logarithmic Derivative
| [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. |
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
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
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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}
}
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 -
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