Neville theta functions

For other θ functions, see Theta function (disambiguation).

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

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

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

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

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

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

Relationship to other functions

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

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).

• $\theta _{c}(2.5,0.3)=-0.65900466676738154967$
• $\theta _{d}(2.5,0.3)=0.95182196661267561994$
• $\theta _{n}(2.5,0.3)=1.0526693354651613637$
• $\theta _{s}(2.5,0.3)=0.82086879524530400536$

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 No such functions in Maple.

References

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
• Neville, E. H. (Eric Harold) (1944). Jacobian Elliptic Functions. Oxford Clarendon Press.
• Weisstein, Eric W. "Neville Theta Functions". MathWorld.