Neville theta functions


In mathematics, the Neville theta functions, named after Eric Harold Neville, are defined as follows:
where: K is the complete elliptic integral of the first kind, K'=K, and is the elliptic nome.
Note that the functions θp are sometimes defined in terms of the nome q and written θp. The functions may also be written in terms of the τ parameter θp where.

Relationship to other functions

The Neville theta functions may be expressed in terms of the Jacobi theta functions
where.
The Neville theta functions are related to the Jacobi elliptic functions. If pq is a Jacobi elliptic function, then

Examples

Substitute z = 2.5, m = 0.3 into the above definitions of Neville theta functions once obtain the following.

Symmetry

Complex 3D plots

Implementation

NetvilleThetaC, NevilleThetaD, NevilleThetaN, and NevilleThetaS are built-in functions of Mathematica
No such functions in Maple.