A new proof of the transformation law of Jacobi’s theta function $\theta _3(w,\tau )$
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- by Wissam Raji
- Proc. Amer. Math. Soc. 135 (2007), 3127-3132
- DOI: https://doi.org/10.1090/S0002-9939-07-08867-3
- Published electronically: June 21, 2007
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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
- 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, DOI 10.1007/978-1-4612-0999-7
- Bruce C. Berndt, Ramanujan’s notebooks. Part III, Springer-Verlag, New York, 1991. MR 1117903, DOI 10.1007/978-1-4612-0965-2
- Kongsiriwong S. A generalization of Siegel’s method, submitted for publication.
- Hans Rademacher, On the transformation of $\log \eta (\tau )$, J. Indian Math. Soc. (N.S.) 19 (1955), 25–30. MR 70660
- Carl Ludwig Siegel, A simple proof of $\eta (-1/\tau )=\eta (\tau )\sqrt {}\tau /i$, Mathematika 1 (1954), 4. MR 62774, DOI 10.1112/S0025579300000462
- Whittaker E.T. and Watson G.N., A course in Modern Analysis, Cambridge Mathematical Library, U.K., 2002.
Bibliographic Information
- Wissam Raji
- Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
- Email: wissam@temple.edu
- 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
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 3127-3132
- MSC (2000): Primary 11F11, 11F99
- DOI: https://doi.org/10.1090/S0002-9939-07-08867-3
- MathSciNet review: 2322742