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).
Symmetry
Complex 3D plots
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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
- ^ Abramowitz and Stegun, pp. 578-579
- ^ Neville (1944)
- ^ wolfram Mathematic
- ^ wolfram math
- ^ 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.
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: Explicit use of et al. in:|editor2-last=
(help) - ^ [1]
- ^ [2]
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.