In mathematics, the Neville theta functions, named after Eric Harold Neville,[1] are defined as follows:[2][3] [4]
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23c24bd9586d43e40359544742532cde9e54542f)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d839d12fde64c9c4137ad8884f6823b3517f5003)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b54ebbb8587e676aac5348e5d2879dddfd0dcf9)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/886ed86bc171fc62c4d5e250ff58b4afeef6bebb)
where: K(m) is the complete elliptic integral of the first kind,
, and
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[5]). The functions may also be written in terms of the τ parameter θp(z|τ) where
.
Relationship to other functions
The Neville theta functions may be expressed in terms of the Jacobi theta functions[5]
![{\displaystyle \theta _{s}(z|\tau )=\theta _{3}^{2}(0|\tau )\theta _{1}(z'|\tau )/\theta '_{1}(0|\tau )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7fbcbf1246874c6d480344cdd7ae1ef10fb08f)
![{\displaystyle \theta _{c}(z|\tau )=\theta _{2}(z'|\tau )/\theta _{2}(0|\tau )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/778f6cc79cad07a0eacf683ddd41133013dd0513)
![{\displaystyle \theta _{n}(z|\tau )=\theta _{4}(z'|\tau )/\theta _{4}(0|\tau )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfa50167c830b53a3e2df2484ce4877bbbadce71)
![{\displaystyle \theta _{d}(z|\tau )=\theta _{3}(z'|\tau )/\theta _{3}(0|\tau )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d4ba7c0a136e62fbda52788f5cb586ead6c546)
where
.
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
![{\displaystyle \operatorname {pq} (u,m)={\frac {\theta _{p}(u,m)}{\theta _{q}(u,m)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b8112efd7ceff4e1cd94535a21ea1da5b0d1cf)
Examples
![{\displaystyle \theta _{c}(2.5,0.3)\approx -0.65900466676738154967}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccad4b9e261a3a11d376fd50bbcef50db0c4090)
![{\displaystyle \theta _{d}(2.5,0.3)\approx 0.95182196661267561994}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02c0a90253b192730a47cc7a443dbd6d11dff62d)
![{\displaystyle \theta _{n}(2.5,0.3)\approx 1.0526693354651613637}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60447a8bbfb6bb6d5091e86cc45a0cc1d54ac432)
![{\displaystyle \theta _{s}(2.5,0.3)\approx 0.82086879524530400536}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77c9ce4f83db3c958ade4f232cdaa3910cd21d50)
Symmetry
![{\displaystyle \theta _{c}(z,m)=\theta _{c}(-z,m)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24cdb73ab22d0e6459105986101238a6d7a3e76d)
![{\displaystyle \theta _{d}(z,m)=\theta _{d}(-z,m)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6046d928c3fba6f7c585921b6afc17c9629b3844)
![{\displaystyle \theta _{n}(z,m)=\theta _{n}(-z,m)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6174d0eb8c973dbccd9a6418e077f908c189840f)
![{\displaystyle \theta _{s}(z,m)=-\theta _{s}(-z,m)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c82ce4715a0003a7349e31f1713c32393a52d1da)
Complex 3D plots
Notes
- ^ Abramowitz and Stegun, pp. 578-579
- ^ Neville (1944)
- ^ The Mathematical Functions Site
- ^ The Mathematical Functions Site
- ^ a b 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.
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.