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

Jacobi Theta Functions (Wolfram MathWorld)
Elliptic Functions (The Wolfram Functions Site)
EllipticTheta (Wolfram Documentation Center)
EllipticThetaPrime (Wolfram Documentation Center)
Nome (Wolfram MathWorld)
After 100 Years, Ramanujan Gap Filled (Wolfram Blog)

Permanent Citation

Enrique Zeleny
​
​"Jacobi Theta Functions"​
​http://demonstrations.wolfram.com/JacobiThetaFunctions/​
​Wolfram Demonstrations Project​
​Published: January 1, 1999