In mathematics, the Neville theta functions, named after Eric Harold Neville, are defined as follows:

θ c ( z , m ) = 2 π q ( m ) 1 / 4 m 1 / 4 K ( m ) ∑ k = 0 ∞ ( q ( m ) ) k ( k + 1 ) cos ⁡ ( ( 2 k + 1 ) π z 2 K ( m ) ) {\displaystyle \theta _{c}(z,m)={\frac {{\sqrt {2\pi }}\,q(m)^{1/4}}{m^{1/4}{\sqrt {K(m)}}}}\,\,\sum _{k=0}^{\infty }(q(m))^{k(k+1)}\cos \left({\frac {(2k+1)\pi z}{2K(m)}}\right)} θ d ( z , m ) = 2 π 2 K ( m ) ( 1 + 2 ∑ k = 1 ∞ ( q ( m ) ) k 2 cos ⁡ ( π z k K ( m ) ) ) {\displaystyle \theta _{d}(z,m)={\frac {\sqrt {2\pi }}{2{\sqrt {K(m)}}}}\,\,\left(1+2\,\sum _{k=1}^{\infty }(q(m))^{k^{2}}\cos \left({\frac {\pi zk}{K(m)}}\right)\right)} θ n ( z , m ) = 2 π 2 ( 1 − m ) 1 / 4 K ( m ) ( 1 + 2 ∑ k = 1 ∞ ( − 1 ) k ( q ( m ) ) k 2 cos ⁡ ( π z k K ( m ) ) ) {\displaystyle \theta _{n}(z,m)={\frac {\sqrt {2\pi }}{2(1-m)^{1/4}{\sqrt {K(m)}}}}\,\,\left(1+2\sum _{k=1}^{\infty }(-1)^{k}(q(m))^{k^{2}}\cos \left({\frac {\pi zk}{K(m)}}\right)\right)} θ s ( z , m ) = 2 π q ( m ) 1 / 4 m 1 / 4 ( 1 − m ) 1 / 4 K ( m ) ∑ k = 0 ∞ ( − 1 ) k ( q ( m ) ) k ( k + 1 ) sin ⁡ ( ( 2 k + 1 ) π z 2 K ( m ) ) {\displaystyle \theta _{s}(z,m)={\frac {{\sqrt {2\pi }}\,q(m)^{1/4}}{m^{1/4}(1-m)^{1/4}{\sqrt {K(m)}}}}\,\,\sum _{k=0}^{\infty }(-1)^{k}(q(m))^{k(k+1)}\sin \left({\frac {(2k+1)\pi z}{2K(m)}}\right)}

where: K(m) is the complete elliptic integral of the first kind, K ′ ( m ) = K ( 1 − m ) {\displaystyle K'(m)=K(1-m)} , and q ( m ) = e − π K ′ ( m ) / K ( m ) {\displaystyle q(m)=e^{-\pi K'(m)/K(m)}} is the elliptic nome.

Note that the functions θp(z,m) are sometimes defined in terms of the nome q(m) and written θp(z,q) (e.g. NIST). The functions may also be written in terms of the τ parameter θp(z|τ) where q = e i π τ {\displaystyle q=e^{i\pi \tau }} .