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

Author(s): Wissam Raji
Journal: Proc. Amer. Math. Soc. 135 (2007), 3127-3132.
MSC (2000): Primary 11F11, 11F99
Posted: 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:

1.: Apostol, T. Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1989. MR 1027834 (90j:11001)
2.: Berndt, Bruce Ramanujan's Notebooks: Part III, Springer-Verlag, New York, 1991. MR 1117903 (92j:01069)
3.: Kongsiriwong S. A generalization of Siegel's method, submitted for publication.
4.: Rademacher, H. On the transformation of $\log \eta(\tau)$ . J. Indian Math Soc. 19(1955), 25-30. MR 0070660 (17:15f)
5.: Siegel, C. A simple proof of $\eta(-\frac{1}{\tau})=\eta(\tau)\sqrt{\frac{\tau}{i}}$ , J. Mathematika 1(1954), 4. MR 0062774 (16:16b)
6.: Whittaker E.T. and Watson G.N., A course in Modern Analysis, Cambridge Mathematical Library, U.K., 2002.

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Additional Information:

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

DOI: 10.1090/S0002-9939-07-08867-3
PII: S 0002-9939(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
Posted: June 21, 2007
Communicated by: Wen-Ching Winnie Li
Copyright of article: Copyright 2007, American Mathematical Society


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