Jacobi Theta Functions

Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
Theta functions are a family of special functions , important in number theory, analysis, heat conduction, representation of solitons, and quantum field theory. The plots show theta functions for complex
and nome (a parameter used for elliptic functions)
(left) and for complex
on the unit disk (right). The black dot in the graphic on the right indicates the point
. These functions are related to several other special functions: the Dedekind
function, the Weierstrass elliptic functions, and the Riemann zeta function, with many identities connecting them [1, 2]. For a basic introduction to elliptic functions, see [3]; generalizations of theta functions include the Ramanujan theta functions.
Contributed by: Enrique Zeleny (October 2014)
Open content licensed under CC BY-NC-SA
Snapshots
Details
References
[1] Wikipedia. "Theta Function." (Oct 6, 2014) en.wikipedia.org/wiki/Theta_function.
[2] W. P. Reinhardt and P. L. Walker, "Chapter 20: Theta Functions," NIST Digital Library of Mathematical Functions, Release 1.0.9 of 2014-08-29. dlmf.nist.gov/20.
[3] V. G. Tkachev. "Elliptic Functions: Introduction Course." (Nov 7, 2014) http://www.researchgate.net/publication/255655268_Elliptic_functions_Introduction_course.
[4] Souichiro-Ikebe. "Theta function." Graphics Library of Special Functions (in Japanese). (Oct 8, 2015) math-functions-1.watson.jp/sub1_spec_100.html.
Permanent Citation
"Jacobi Theta Functions"
http://demonstrations.wolfram.com/JacobiThetaFunctions/
Wolfram Demonstrations Project
Published: October 7 2014