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.