Automatic reduction of elliptic integrals using Carlson's relations

Author:
Norman Gray

Journal:
Math. Comp. **71** (2002), 311-318

MSC (2000):
Primary 33E05; Secondary 33-04

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

Published electronically:
July 22, 2001

MathSciNet review:
1863002

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Abstract | References | Similar Articles | Additional Information

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.

**[Car87]**B. C. Carlson,*A table of elliptic integrals of the second kind*, Mathematics of Computation**49**(1987), no. 180, 595-606. MR**89b:65013****[Car88]**B. C. Carlson,*A table of elliptic integrals of the third kind*, Mathematics of Computation**51**(1988), no. 183, 267-280. MR**89k:33003****[Car89]**B. C. Carlson,*A table of elliptic integrals: Cubic cases*, Mathematics of Computation**53**(1989), no. 187, 327-333. MR**89m:65009****[Car91]**B. C. Carlson,*A table of elliptic integrals: One quadratic factor*, Mathematics of Computation**56**(1991), no. 193, 267-280. MR**92b:33056****[Car98]**B. C. Carlson,*Elliptic integrals: Symmetry and symbolic integration*, Atti dei Convengni Lincei**147**(1998), 161-181. MR**2001f:33024****[Car99]**B. C. Carlson,*Toward symbolic integration of elliptic integrals*, J. Symbolic Computation**28**(1999), no. 6, 739-753. MR**2001f:33025****[GR94]**I. S. Gradshteyn and I. M. Ryzhik,*Table of integrals, series, and products*, 5th ed., Academic Press, 1994. MR**94g:00008****[NP76]**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.

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Additional Information

**Norman Gray**

Affiliation:
Department of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, United Kingdom

Email:
norman@astro.gla.ac.uk

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

Keywords:
Carlson elliptic integrals

Received by editor(s):
August 16, 1999

Received by editor(s) in revised form:
May 5, 2000

Published electronically:
July 22, 2001

Article copyright:
© Copyright 2001
American Mathematical Society