Neville theta functions

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

$θ_{c} (z, m) = \frac{\sqrt{2 π} q (m)^{1 / 4}}{m^{1 / 4} \sqrt{K (m)}} \sum_{k = 0}^{\infty} (q (m))^{k (k + 1)} \cos (\frac{(2 k + 1) π z}{2 K (m)})$

$θ_{d} (z, m) = \frac{\sqrt{2 π}}{2 \sqrt{K (m)}} (1 + 2 \sum_{k = 1}^{\infty} (q (m))^{k^{2}} \cos (\frac{π z k}{K (m)}))$

$θ_{n} (z, m) = \frac{\sqrt{2 π}}{2 (1 - m)^{1 / 4} \sqrt{K (m)}} (1 + 2 \sum_{k = 1}^{\infty} (- 1)^{k} (q (m))^{k^{2}} \cos (\frac{π z k}{K (m)}))$

$θ_{s} (z, m) = \frac{\sqrt{2 π} q (m)^{1 / 4}}{m^{1 / 4} (1 - m)^{1 / 4} \sqrt{K (m)}} \sum_{k = 0}^{\infty} (- 1)^{k} (q (m))^{k (k + 1)} \sin (\frac{(2 k + 1) π z}{2 K (m)})$

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

Note that the functions θ_p(z,m) are sometimes defined in terms of the nome q(m) and written θ_p(z,q) (e.g. NIST^[5]). The functions may also be written in terms of the τ parameter θ_p(z|τ) where $q = e^{i π τ}$ .

Relationship to other functions

The Neville theta functions may be expressed in terms of the Jacobi theta functions^[5]

$θ_{s} (z | τ) = θ_{3}^{2} (0 | τ) θ_{1} (z^{'} | τ) / θ'_{1} (0 | τ)$ $θ_{c} (z | τ) = θ_{2} (z^{'} | τ) / θ_{2} (0 | τ)$ $θ_{n} (z | τ) = θ_{4} (z^{'} | τ) / θ_{4} (0 | τ)$ $θ_{d} (z | τ) = θ_{3} (z^{'} | τ) / θ_{3} (0 | τ)$

where $z^{'} = z / θ_{3}^{2} (0 | τ)$ .

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

$pq (u, m) = \frac{θ_{p} (u, m)}{θ_{q} (u, m)} .$

Examples

$θ_{c} (2.5, 0.3) \approx - 0.65900466676738154967$
$θ_{d} (2.5, 0.3) \approx 0.95182196661267561994$
$θ_{n} (2.5, 0.3) \approx 1.0526693354651613637$
$θ_{s} (2.5, 0.3) \approx 0.82086879524530400536$

Symmetry

$θ_{c} (z, m) = θ_{c} (- z, m)$
$θ_{d} (z, m) = θ_{d} (- z, m)$
$θ_{n} (z, m) = θ_{n} (- z, m)$
$θ_{s} (z, m) = - θ_{s} (- z, m)$

Complex 3D plots

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. pp. 578–579. 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.
↑ The Mathematical Functions Site
↑ The Mathematical Functions Site
1 2 Olver, F. W. J.; et al., eds. (2017-12-22). "NIST Digital Library of Mathematical Functions (Release 1.0.17)". National Institute of Standards and Technology. Retrieved 2018-02-26.

Weisstein, Eric W. "Neville Theta Functions". MathWorld.

[1] 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. pp. 578–579. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.

[2] Neville, E. H. (Eric Harold) (1944). Jacobian Elliptic Functions. Oxford Clarendon Press.

[3] The Mathematical Functions Site

[4] The Mathematical Functions Site

[DLMF20-5] 1 2 Olver, F. W. J.; et al., eds. (2017-12-22). "NIST Digital Library of Mathematical Functions (Release 1.0.17)". National Institute of Standards and Technology. Retrieved 2018-02-26.

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