Neville theta functions

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

where: K(m) is the complete elliptic integral of the first kind, K'(m)=K(1-m), 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]

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

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).

  • [6]

Symmetry

Complex 3D plots

Implementation

NetvilleThetaC[z,m], NevilleThetaD[z,m], NevilleThetaN[z,m], and NevilleThetaS[z,m] are built-in functions of Mathematica[7] No such functions in Maple.

Notes

  1. Abramowitz and Stegun, pp. 578-579
  2. Neville (1944)
  3. wolfram Mathematic
  4. wolfram math
  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

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