Calculation of the Taylor series expansion coefficients of the Jacobian elliptic function

Author:
Staffan Wrigge

Journal:
Math. Comp. **36** (1981), 555-564

MSC:
Primary 65D20; Secondary 33A25

DOI:
https://doi.org/10.1090/S0025-5718-1981-0606513-8

MathSciNet review:
606513

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Abstract | References | Similar Articles | Additional Information

Abstract: The Taylor series expansion coefficients of the Jacobian elliptic function and its power are studied. Recurrence formulae are given, and tables of the coefficients constructed. Using Lagrange's inversion formula, these coefficients can be expressed in terms of Legendre polynomials.

**[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 1972.**[2]**F. Bowman,*Introduction to Elliptic Functions with Applications*, Dover, New York, 1961. MR**0132214 (24:A2060)****[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]**H. Hancock,*Lectures on the Theory of Elliptic Functions*, Dover, New York, 1958.**[5]**I. S. Gradshteyn & I. M. Ryshik,*Tables of Series, Products and Integrals*, Academic Press, New York, 1965.**[6]**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)****[7]**E. T. Whittaker & G. N. Watson,*Modern Analysis*, Cambridge Univ. Press, Cambridge, 1927.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1981-0606513-8

Keywords:
Special functions,
elliptic functions,
Legendre polynomials

Article copyright:
© Copyright 1981
American Mathematical Society