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Neville theta functions

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

Relationship to other functions
The Neville theta functions may be expressed in terms of the Jacobi theta functions :\theta_s(z|\tau)=\theta_3^2(0|\tau)\theta_1(z'|\tau)/\theta'_1(0|\tau) :\theta_c(z|\tau)=\theta_2(z'|\tau)/\theta_2(0|\tau) :\theta_n(z|\tau)=\theta_4(z'|\tau)/\theta_4(0|\tau) :\theta_d(z|\tau)=\theta_3(z'|\tau)/\theta_3(0|\tau) where z'=z/\theta_3^2(0|\tau). 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 :\operatorname{pq}(u,m)=\frac{\theta_p(u,m)}{\theta_q(u,m)}. ==Examples==
Examples
• \theta_c(2.5, 0.3)\approx -0.65900466676738154967 • \theta_d(2.5, 0.3)\approx 0.95182196661267561994 • \theta_n(2.5, 0.3)\approx 1.0526693354651613637 • \theta_s(2.5, 0.3)\approx 0.82086879524530400536 ==Symmetry==
Symmetry
• \theta_c(z,m)=\theta_c(-z,m) • \theta_d(z,m)=\theta_d(-z,m) • \theta_n(z,m)=\theta_n(-z,m) • \theta_s(z,m)=-\theta_s(-z,m) ==Complex 3D plots==
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