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)

 

 

A new proof of the transformation law of Jacobi's theta function $ \theta_3(w,\tau)$


Author: Wissam Raji
Journal: Proc. Amer. Math. Soc. 135 (2007), 3127-3132
MSC (2000): Primary 11F11, 11F99
Published electronically: June 21, 2007
MathSciNet review: 2322742
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present a new proof, using Residue Calculus, of the transformation law of the Jacobi theta function $ \theta_3(w,\tau)$ defined in the upper half plane. Our proof is inspired by Siegel's proof of the transformation law of the Dedekind eta function.


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

  • 1. Tom M. Apostol, Modular functions and Dirichlet series in number theory, 2nd ed., Graduate Texts in Mathematics, vol. 41, Springer-Verlag, New York, 1990. MR 1027834
  • 2. Bruce C. Berndt, Ramanujan’s notebooks. Part III, Springer-Verlag, New York, 1991. MR 1117903
  • 3. Kongsiriwong S. A generalization of Siegel's method, submitted for publication.
  • 4. Hans Rademacher, On the transformation of log𝜂(𝜏), J. Indian Math. Soc. (N.S.) 19 (1955), 25–30. MR 0070660
  • 5. Carl Ludwig Siegel, A simple proof of 𝜂(-1/𝜏)=𝜂(𝜏)√𝜏/𝑖, Mathematika 1 (1954), 4. MR 0062774
  • 6. Whittaker E.T. and Watson G.N., A course in Modern Analysis, Cambridge Mathematical Library, U.K., 2002.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11F11, 11F99

Retrieve articles in all journals with MSC (2000): 11F11, 11F99


Additional Information

Wissam Raji
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Email: wissam@temple.edu

DOI: https://doi.org/10.1090/S0002-9939-07-08867-3
Keywords: Jacobi theta function, Dedekind eta function, Arzela bounded convergence theorem.
Received by editor(s): February 2, 2006
Received by editor(s) in revised form: July 14, 2006, July 24, 2006, and July 28, 2006
Published electronically: June 21, 2007
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2007 American Mathematical Society