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 ) = 2 π q ( m ) 1 / 4 m 1 / 4 K ( m ) ∑ k = 0 ∞ ( q ( m ) ) k ( k + 1 ) cos ⁡ ( ( 2 k + 1 ) π z 2 K ( m ) ) {\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)}
θ d ( z , m ) = 2 π 2 K ( m ) ( 1 + 2 ∑ k = 1 ∞ ( q ( m ) ) k 2 cos ⁡ ( π z k K ( m ) ) ) {\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)}
θ n ( z , m ) = 2 π 2 ( 1 − m ) 1 / 4 K ( m ) ( 1 + 2 ∑ k = 1 ∞ ( − 1 ) k ( q ( m ) ) k 2 cos ⁡ ( π z k K ( m ) ) ) {\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)}
θ s ( z , m ) = 2 π q ( m ) 1 / 4 m 1 / 4 ( 1 − m ) 1 / 4 K ( m ) ∑ k = 0 ∞ ( − 1 ) k ( q ( m ) ) k ( k + 1 ) sin ⁡ ( ( 2 k + 1 ) π z 2 K ( m ) ) {\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)}

where: K(m) is the complete elliptic integral of the first kind, K ′ ( m ) = K ( 1 − m ) {\displaystyle K'(m)=K(1-m)} , and q ( m ) = e − π K ′ ( m ) / K ( m ) {\displaystyle q(m)=e^{-\pi K'(m)/K(m)}} 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 q = e i π τ {\displaystyle q=e^{i\pi \tau }} .

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 ) {\displaystyle \theta _{s}(z|\tau )=\theta _{3}^{2}(0|\tau )\theta _{1}(z'|\tau )/\theta '_{1}(0|\tau )}
θ c ( z | τ ) = θ 2 ( z ′ | τ ) / θ 2 ( 0 | τ ) {\displaystyle \theta _{c}(z|\tau )=\theta _{2}(z'|\tau )/\theta _{2}(0|\tau )}
θ n ( z | τ ) = θ 4 ( z ′ | τ ) / θ 4 ( 0 | τ ) {\displaystyle \theta _{n}(z|\tau )=\theta _{4}(z'|\tau )/\theta _{4}(0|\tau )}
θ d ( z | τ ) = θ 3 ( z ′ | τ ) / θ 3 ( 0 | τ ) {\displaystyle \theta _{d}(z|\tau )=\theta _{3}(z'|\tau )/\theta _{3}(0|\tau )}

where z ′ = z / θ 3 2 ( 0 | τ ) {\displaystyle 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

pq ⁡ ( u , m ) = θ p ( u , m ) θ q ( u , m ) . {\displaystyle \operatorname {pq} (u,m)={\frac {\theta _{p}(u,m)}{\theta _{q}(u,m)}}.}

Examples

  • θ c ( 2.5 , 0.3 ) ≈ − 0.65900466676738154967 {\displaystyle \theta _{c}(2.5,0.3)\approx -0.65900466676738154967}
  • θ d ( 2.5 , 0.3 ) ≈ 0.95182196661267561994 {\displaystyle \theta _{d}(2.5,0.3)\approx 0.95182196661267561994}
  • θ n ( 2.5 , 0.3 ) ≈ 1.0526693354651613637 {\displaystyle \theta _{n}(2.5,0.3)\approx 1.0526693354651613637}
  • θ s ( 2.5 , 0.3 ) ≈ 0.82086879524530400536 {\displaystyle \theta _{s}(2.5,0.3)\approx 0.82086879524530400536}

Symmetry

  • θ c ( z , m ) = θ c ( − z , m ) {\displaystyle \theta _{c}(z,m)=\theta _{c}(-z,m)}
  • θ d ( z , m ) = θ d ( − z , m ) {\displaystyle \theta _{d}(z,m)=\theta _{d}(-z,m)}
  • θ n ( z , m ) = θ n ( − z , m ) {\displaystyle \theta _{n}(z,m)=\theta _{n}(-z,m)}
  • θ s ( z , m ) = − θ s ( − z , m ) {\displaystyle \theta _{s}(z,m)=-\theta _{s}(-z,m)}

Complex 3D plots

Notes

  1. ^ Abramowitz and Stegun, pp. 578-579
  2. ^ Neville (1944)
  3. ^ The Mathematical Functions Site
  4. ^ The Mathematical Functions Site
  5. ^ 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.