Carlson, B. C. A table of elliptic integrals of the third kind. (English) Zbl 0647.33001 Math. Comput. 51, No. 183, 267-280 (1988); Supplement S1-S5 (1988). Summary: 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 \(R_ C\) and \(R_ J\), revised to include their Cauchy principal values, are listed in a Supplement. Cited in 12 Documents MSC: 33E05 Elliptic functions and integrals 33C05 Classical hypergeometric functions, \({}_2F_1\) Keywords:elliptic integrals; R-functions Software:algorithm 577 PDFBibTeX XMLCite \textit{B. C. Carlson}, Math. Comput. 51, No. 183, 267--280 (1988; Zbl 0647.33001) Full Text: DOI Digital Library of Mathematical Functions: §19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals References: [1] Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. · Zbl 0213.16602 [2] Billie Chandler Carlson, Special functions of applied mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. [3] B. C. Carlson, Computing elliptic integrals by duplication, Numer. Math. 33 (1979), no. 1, 1 – 16. · Zbl 0438.65029 · doi:10.1007/BF01396491 [4] B. C. Carlson, A table of elliptic integrals of the second kind, Math. Comp. 49 (1987), no. 180, 595 – 606, S13 – S17. · Zbl 0625.33002 [5] B. C. Carlson & Elaine M. Notis, ”ALGORITHM 577, Algorithms for incomplete elliptic integrals,” ACM Trans. Math. Software, v. 7, 1981, pp. 398-403. · Zbl 0464.65008 [6] G. Fubini, ”Nuovo metodo per lo studio e per il calcolo delle funzioni trascendenti elementan,” Period. Mat., v. 12, 1897, pp. 169-178. · JFM 28.0372.02 [7] I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980. · Zbl 0521.33001 [8] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 1, Gordon & Breach Science Publishers, New York, 1986. Elementary functions; Translated from the Russian and with a preface by N. M. Queen. · Zbl 0733.00004 [9] D. G. Zill and B. C. Carlson, Symmetric elliptic integrals of the third kind, Math. Comp. 24 (1970), 199 – 214. · Zbl 0199.50001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.