Fourier series coefficients for powers of the Jacobian elliptic functions

Author:
AyΕe Kiper

Journal:
Math. Comp. **43** (1984), 247-259

MSC:
Primary 33A25

DOI:
https://doi.org/10.1090/S0025-5718-1984-0744934-6

MathSciNet review:
744934

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Fourier series expansion coefficients for the Jacobian elliptic functions , and , with , are studied. Two-term recurrence formulae are obtained and some of the coefficients are tabulated.

**[1]**M. Abramowitz & I. A. Stegun,*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*, Nat. Bur. Standards, Appl. Math. Series No. 55, December 1954. (Also: Dover, New York, 1968.)**[2]**Paul F. Byrd and Morris D. Friedman,*Handbook of elliptic integrals for engineers and scientists*, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. MR**0277773****[3]**Dominique Dumont,*A combinatorial interpretation for the Schett recurrence on the Jacobian elliptic functions*, Math. Comp.**33**(1979), no. 148, 1293–1297. MR**537974**, https://doi.org/10.1090/S0025-5718-1979-0537974-1**[4]**Patrick Du Val,*Elliptic functions and elliptic curves*, Cambridge University Press, London-New York, 1973. London Mathematical Society Lecture Note Series, No. 9. MR**0379512****[5]**Arne Fransén,*Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function 𝑠𝑛(𝑥,𝑘)*, Math. Comp.**37**(1981), no. 156, 475–494. MR**628708**, https://doi.org/10.1090/S0025-5718-1981-0628708-X**[6]**R. G. Langebartel,*Fourier expansions of rational fractions of elliptic integrals and Jacobian elliptic functions*, SIAM J. Math. Anal.**11**(1980), no. 3, 506–513. MR**572201**, https://doi.org/10.1137/0511048**[7]**Alois Schett,*Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions*, Math. Comp.**30**(1976), no. 133, 143–147. MR**0391477**, https://doi.org/10.1090/S0025-5718-1976-0391477-3**[8]**A. Schett, Addendum to "Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions,"*Math. Comp.*, v. 31, 1977, Microfiche supplement.**[9]**Alois Schett,*Recurrence formula of the Taylor series expansion coefficients of the Jacobian elliptic functions*, Math. Comp.**31**(1977), no. 140, 1003–1005. MR**0442301**, https://doi.org/10.1090/S0025-5718-1977-0442301-2**[10]**E. T. Whittaker and G. N. Watson,*A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions*, Fourth edition. Reprinted, Cambridge University Press, New York, 1962. MR**0178117****[11]**Staffan Wrigge,*Calculation of the Taylor series expansion coefficients of the Jacobian elliptic function 𝑠𝑛(𝑥,𝑘)*, Math. Comp.**36**(1981), no. 154, 555–564. MR**606513**, https://doi.org/10.1090/S0025-5718-1981-0606513-8**[12]**Staffan Wrigge,*A note on the Taylor series expansion coefficients of the Jacobian elliptic function 𝑠𝑛(𝑥,𝑘)*, Math. Comp.**37**(1981), no. 156, 495–497. MR**628709**, https://doi.org/10.1090/S0025-5718-1981-0628709-1

Retrieve articles in *Mathematics of Computation*
with MSC:
33A25

Retrieve articles in all journals with MSC: 33A25

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1984-0744934-6

Article copyright:
© Copyright 1984
American Mathematical Society