A table of elliptic integrals of the second kind
Author:
B. C. Carlson
Journal:
Math. Comp. 49 (1987), 595606, S13
MSC:
Primary 65A05; Secondary 33A25, 65V05
MathSciNet review:
906192
Fulltext PDF Free Access
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Abstract: By evaluating elliptic integrals in terms of standard Rfunctions instead of Legendre's integrals, many (in one case 144) formulas in previous tables are unified. The present table includes only integrals of the first and second kinds having integrands with real singular points. The 216 integrals of this type listed in Gradshteyn and Ryzhik's table constitute a small fraction of the special cases of 13 integrals evaluated here. The interval of integration is not required, as it is in previous tables, to begin or end at a singular point of the integrand. Fortran codes for the standard Rfunctions are included in a Supplement.
 [1]
Paul
F. Byrd and Morris
D. Friedman, Handbook of elliptic integrals for engineers and
scientists, Die Grundlehren der mathematischen Wissenschaften, Band
67, SpringerVerlag, New YorkHeidelberg, 1971. Second edition, revised. MR 0277773
(43 #3506)
 [2]
Billie
Chandler Carlson, Special functions of applied mathematics,
Academic Press [Harcourt Brace Jovanovich, Publishers], New YorkLondon,
1977. MR
0590943 (58 #28707)
 [3]
B.
C. Carlson, Elliptic integrals of the first kind, SIAM J.
Math. Anal. 8 (1977), no. 2, 231–242. MR 0430341
(55 #3346)
 [4]
B.
C. Carlson, Short proofs of three theorems on elliptic
integrals, SIAM J. Math. Anal. 9 (1978), no. 3,
524–528. MR 0473247
(57 #12922)
 [5]
B.
C. Carlson, Computing elliptic integrals by duplication,
Numer. Math. 33 (1979), no. 1, 1–16. MR 545738
(80h:65008), http://dx.doi.org/10.1007/BF01396491
 [6]
B. C. Carlson & Elaine M. Notis, "ALGORITHM 577, Algorithms for incomplete elliptic integrals," ACM Trans. Math. Software, v. 7, 1981, pp. 398403.
 [7]
I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980.
 [8]
D. H. Lehmer, "The lemniscate constant," Math. Comp., v. 3, 1948/49, pp. 550551.
 [9]
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. MR 874986
(88f:00013)
 [10]
John
Todd, The lemniscate constants, Comm. ACM 18
(1975), 14–19; corrigendum, ibid. 18 (1975), no. 8, 462. Collection
of articles honoring Alston S. Householder. MR 0375745
(51 #11935)
 [11]
D.
G. Zill and B.
C. Carlson, Symmetric elliptic integrals of the
third kind, Math. Comp. 24 (1970), 199–214. MR 0262553
(41 #7159), http://dx.doi.org/10.1090/S00255718197002625535
 [1]
 P. F. Byrd & M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed., SpringerVerlag, 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, "Elliptic integrals of the first kind," SIAM J. Math. Anal., v. 8, 1977, pp. 231242. MR 0430341 (55:3346)
 [4]
 B. C. Carlson, "Short proofs of three theorems on elliptic integrals," SIAM J. Math. Anal., v. 9, 1978, pp. 524528. MR 0473247 (57:12922)
 [5]
 B. C. Carlson, "Computing elliptic integrals by duplication," Numer. Math., v. 33, 1979, pp. 116. MR 545738 (80h:65008)
 [6]
 B. C. Carlson & Elaine M. Notis, "ALGORITHM 577, Algorithms for incomplete elliptic integrals," ACM Trans. Math. Software, v. 7, 1981, pp. 398403.
 [7]
 I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980.
 [8]
 D. H. Lehmer, "The lemniscate constant," Math. Comp., v. 3, 1948/49, pp. 550551.
 [9]
 A. P. Prudnikov, Yu. A. Brychkov & O. I. Marichev, Integrals and Series, vol. 1, Gordon and Breach, New York, 1986. MR 874986 (88f:00013)
 [10]
 John Todd, "The lemniscate constants," Comm. ACM, v. 18, 1975, pp. 1419, 462. MR 0375745 (51:11935)
 [11]
 D. G. Zill & B. C. Carlson, "Symmetric elliptic integrals of the third kind," Math. Comp., v. 24, 1970, pp. 199214. MR 0262553 (41:7159)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198709061921
PII:
S 00255718(1987)09061921
Article copyright:
© Copyright 1987
American Mathematical Society
