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Fourier series coefficients for powers of the Jacobian elliptic functions


Author: Ayşe 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
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Abstract: The Fourier series expansion coefficients for the Jacobian elliptic functions $ {\text{sn}^m}(u,k)$, $ {\text{cn}^m}(u,k)$ and $ {\text{dn}^m}(u,k)$, with $ m \geqslant 1$, are studied. Two-term recurrence formulae are obtained and some of the coefficients are tabulated.


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  • [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] P. F. Byrd & M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed., Springer-Verlag, Berlin, 1971. MR 0277773 (43:3506)
  • [3] D. Dumont, "A combinatorial interpretation for the Schett recurrence on the Jacobian elliptic functions," Math. Comp., v. 33, 1979, pp. 1293-1297. MR 537974 (80i:33003)
  • [4] P. Du Val, Elliptic Functions and Elliptic Curves, London Mathematical Society Lecture Note Series 9, Cambridge Univ. Press, Cambridge, 1973. MR 0379512 (52:417)
  • [5] A. Fransén, "Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function $ {\text{sn}}(x,k)$," Math. Comp., v. 37, 1981, pp. 475-494. MR 628708 (83a:33003)
  • [6] R. G. Langebartel, "Fourier expansions of rational fractions of elliptic integrals and Jacobian elliptic functions," SIAM J. Math. Anal., v. 11, 1980, pp. 506-513. MR 572201 (81e:42008)
  • [7] A. Schett, "Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions," Math. Comp., v. 30, 1976, pp. 143-147. MR 0391477 (52:12298)
  • [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] A. Schett, "Recurrence formula of the Taylor series expansion coefficients of the Jacobian elliptic functions," Math. Comp., v. 31, 1977, pp. 1003-1005. MR 0442301 (56:687)
  • [10] E. T. Whittaker & G. N. Watson, Modern Analysis, Cambridge Univ. Press, Cambridge, 1962. MR 0178117 (31:2375)
  • [11] S. Wrigge, "Calculation of the Taylor series expansion coefficients of the Jacobian elliptic function $ {\text{sn}}(x,k)$," Math. Comp., v. 36, 1981, pp. 555-564. MR 606513 (82d:65023)
  • [12] S. Wrigge, "A note on the Taylor series expansion coefficients of the Jacobian elliptic function $ {\text{sn}}(x,k)$," Math. Comp., v. 37, 1981, pp. 495-497. MR 628709 (83a:33004)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1984-0744934-6
Article copyright: © Copyright 1984 American Mathematical Society

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