Misplaced Pages

Neville theta functions

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
For other θ functions, see Theta function (disambiguation).

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

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). 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

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. ^ 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

Categories: