Neville theta functions

In mathematics, the Neville theta functions, named after Eric Harold Neville,^[1] are defined as follows:^{[2][3] [4]}

$\theta_c(z,m)=\sqrt {2}\sqrt {\pi }\sqrt [4]{q(m)} \sum _{k=0}^\infty (q(m))^{k(k+1)} \cos \left( \frac 1 2 \cdot \frac {( 2k+1) \pi z}{K(m)} \right) \frac {1}{\sqrt {K(m)}} \frac {1}{\sqrt [4]{m}}$

$\theta_d(z,m)=1/2\,\sqrt {2}\sqrt {\pi } \left( 1+2\,\sum _{k=1}^\infty (q(m))^{k^2} \cos \left( \frac {\pi zk}{K(m)} \right) \right) \frac {1}{\sqrt {K(m)}}$

$\theta_n(z, m) = 1/2\,\sqrt {\pi } \sqrt {2} \left( 1+2\sum _{k=1}^\infty (-1)^k (q(m))^{k^2} \cos \left( \frac {\pi zk}{K(m)} \right) \right) \frac {1}{\sqrt [4]{1-m}} \frac {1}{\sqrt {K(m)}}$

$\theta_s(z, m)=\sqrt {\pi }\sqrt {2} \sqrt [4]{q(m)} \sum _{k=0}^\infty (-1)^k (q(m))^{k(k+1) } \sin \left( 1/2\,\frac { (2k+1) \pi z}{K(m)} \right) \frac {1}{\sqrt [4]{1-m}} \frac{1}{\sqrt [4]{m}} \frac {1}{\sqrt {K(m)}}$

where:

• $K(m)=\operatorname{EllipticK}(\sqrt m)$
• $K'(m)=\operatorname{EllipticK}(\sqrt{1-m})$
• $q(m)=e^{-\pi K(m)/K'(m)}$ is the elliptic nome

Examples

Substitute z = 2.5, m = 0.3 into the above definitions of Neville theta functions(using Maple) once obtain the following(consistent with results from wolfram math).

• $\theta_c(2.5, 0.3)=-0.65900466676738154967$^[5]
• $\theta_d(2.5, 0.3)=0.95182196661267561994$
• $\theta_n(2.5, 0.3)=1.0526693354651613637$
• $\theta_s(2.5, 0.3)=0.82086879524530400536$

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

Implementation

NetvilleThetaC[z,m], NevilleThetaD[z,m], NevilleThetaN[z,m], and NevilleThetaS[z,m] are built-in functions of Mathematica^[6] No such functions in Maple.

Notes

1. Abramowitz and Stegun, pp. 578-579
2. Neville (1944)
3. wolfram Mathematic
4. wolfram math

References

• Abramowitz, Milton; Stegun, Irene A., eds. (December 1972) [1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55 (10 ed.). New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642.
• Neville, E. H. (Eric Harold) (1944). Jacobian Elliptic Functions. Oxford Clarendon Press.
• Weisstein, Eric W., "Neville Theta Functions", MathWorld.