In mathematics, the Neville theta functions, named after Eric Harold Neville, are defined as follows:
where: K(m) is the complete elliptic integral of the first kind, , and is the elliptic nome.
Note that the functions θ<sub>p</sub>(z,m) are sometimes defined in terms of the nome q(m) and written θ<sub>p</sub>(z,q) (e.g. NIST). The functions may also be written in terms of the τ parameter θ<sub>p</sub>(z|τ) where .
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(u,m) is a Jacobi elliptic function (p and q are one of s,c,n,d), then