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Jacobi Theta Functions

​
function
EllipticTheta
EllipticThetaPrime
a
2
u
2
r
0.745
θ
1.045
f
Re
Im
Abs
Arg
Theta functions are a family of special functions
ϑ
a
(u,q)(a=1,…,4)
, important in number theory, analysis, heat conduction, representation of solitons, and quantum field theory. The plots show theta functions for complex
u
and nome (a parameter used for elliptic functions)
q=
iπτ
e
(left) and for complex
q
on the unit disk (right). The black dot in the graphic on the right indicates the point
(r,θ)
. These functions are related to several other special functions: the Dedekind
η
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.

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.

External Links

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