A table of elliptic integrals of the third kind

Author:
B. C. Carlson

Journal:
Math. Comp. **51** (1988), 267-280, S1

MSC:
Primary 33A25; Secondary 65A05

DOI:
https://doi.org/10.1090/S0025-5718-1988-0942154-7

MathSciNet review:
942154

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: As many as 72 elliptic integrals of the third kind in previous tables are unified by evaluation in terms of *R*-functions instead of Legendre's integrals. The present table includes only integrals having integrands with real singular points. In addition to 31 integrals of the third kind, most of them unavailable elsewhere, 10 integrals of the first and second kinds from an earlier table are listed again in new notation. In contrast to conventional tables, the interval of integration is not required to begin or end at a singular point of the integrand. Fortran codes for the standard *R*-functions and , revised to include their Cauchy principal values, are listed in a Supplement.

**[1]**P. F. Byrd & M. D. Friedman,*Handbook of Elliptic Integrals for Engineers and Scientists*, 2nd ed., Springer-Verlag, New York, 1971. MR**0277773 (43:3506)****[2]**B. C. Carlson,*Special Functions of Applied Mathematics*, Academic Press, New York, 1977. MR**0590943 (58:28707)****[3]**B. C. Carlson, "Computing elliptic integrals by duplication,"*Numer. Math.*, v. 33, 1979, pp. 1-16. MR**545738 (80h:65008)****[4]**B. C. Carlson, "A table of elliptic integrals of the second kind,"*Math. Comp.*, v. 49, 1987, pp. 595-606. (Supplement,*ibid.*, S13-S17.) MR**906192 (89b:65013)****[5]**B. C. Carlson & Elaine M. Notis, "ALGORITHM 577, Algorithms for incomplete elliptic integrals,"*ACM Trans. Math. Software*, v. 7, 1981, pp. 398-403.**[6]**G. Fubini, "Nuovo metodo per lo studio e per il calcolo delle funzioni trascendenti elementan,"*Period. Mat.*, v. 12, 1897, pp. 169-178.**[7]**I. S. Gradshteyn & I. M. Ryzhik,*Table of Integrals, Series, and Products*, Academic Press, New York, 1980.**[8]**A. P. Prudnikov, Yu. A. Bbychkov & O. I. Marichev,*Integrals and Series*, Vol. 1, Gordon and Breach, New York, 1986. MR**874986 (88f:00013)****[9]**D. G. Zill & B. C. Carlson, "Symmetric elliptic integrals of the third kind,"*Math. Comp.*, v. 24, 1970, pp. 199-214. MR**0262553 (41:7159)**

Retrieve articles in *Mathematics of Computation*
with MSC:
33A25,
65A05

Retrieve articles in all journals with MSC: 33A25, 65A05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0942154-7

Article copyright:
© Copyright 1988
American Mathematical Society