Automatic reduction of elliptic integrals using Carlson’s relations

Gray, Norman

doi:10.1090/S0025-5718-01-01333-3

Automatic reduction of elliptic integrals using Carlson’s relations
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by Norman Gray;

Math. Comp. 71 (2002), 311-318

DOI: https://doi.org/10.1090/S0025-5718-01-01333-3

Published electronically: July 22, 2001

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Abstract:

In a series of papers, B. C. Carlson produced tables of elliptic integrals, evaluating them in terms of easily computed symmetrical functions, using a group of multivariate recurrence relations. These relations are, however, cumbersome to use by hand and, in the absence of a specific reductive algorithm, difficult to use with computer algebra. This paper presents such an algorithm, guaranteed to reduce a general elliptic integral to a set of fundamental ones.

References

B. C. Carlson, A table of elliptic integrals of the second kind, Math. Comp. 49 (1987), no. 180, 595–606, S13–S17. MR 906192, DOI 10.1090/S0025-5718-1987-0906192-1
B. C. Carlson, A table of elliptic integrals of the third kind, Math. Comp. 51 (1988), no. 183, 267–280, S1–S5. MR 942154, DOI 10.1090/S0025-5718-1988-0942154-7
B. C. Carlson, A table of elliptic integrals: cubic cases, Math. Comp. 53 (1989), no. 187, 327–333. MR 969482, DOI 10.1090/S0025-5718-1989-0969482-4
B. C. Carlson, A table of elliptic integrals: one quadratic factor, Math. Comp. 56 (1991), no. 193, 267–280. MR 1052087, DOI 10.1090/S0025-5718-1991-1052087-6
Bille Chandler Carlson, Elliptic integrals: symmetry and symbolic integration, Tricomi’s ideas and contemporary applied mathematics (Rome/Turin, 1997) Atti Convegni Lincei, vol. 147, Accad. Naz. Lincei, Rome, 1998, pp. 161–181. MR 1737495
B. C. Carlson, Toward symbolic integration of elliptic integrals, J. Symbolic Comput. 28 (1999), no. 6, 739–753. Orthogonal polynomials and computer algebra. MR 1750543, DOI 10.1006/jsco.1999.0336
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 5th ed., Academic Press, Inc., Boston, MA, 1994. Translation edited and with a preface by Alan Jeffrey. MR 1243179
Edward W. Ng and Desanka Polajnar, A study of alternative methods for the symbolic calculation of elliptic integrals, Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, 1976, pp. 372–376.