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Neville theta functions

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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, , 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

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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

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Symmetry

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Complex 3D plots

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Notes

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  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

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