×

Series expansions of symmetric elliptic integrals. (English) Zbl 1245.33019

In order to improve the speed of convergence of series expansions of Carlson’s symmetric elliptic integrals (see, e.g. Chapter 19 of [F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST handbook of mathematical functions. Cambridge: Cambridge University Press (2010; Zbl 1198.00002)]) the author develops several new series expansions by using the symmetric nature of Carlson’s basic integrals. First, he provides 15 series expansions by a new approach. Eleven of these seem to be new. Next, it is discussed the effectiveness of these expansions and their fifteen complements, obtained from the introduction of special addition formulas. Further, it is shown an effective approach to compute the complete elliptic integrals using incomplete ones. Finally, some results of numerical experiments are illustrated. At the end of the paper, three appendices are added, too.

MSC:

33E05 Elliptic functions and integrals

Citations:

Zbl 1198.00002

Software:

algorithm 577; DLMF; NSWC
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abramowitz, M., & Stegun, I.A. , Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Chapter 17. National Bureau of Standards, Washington (1964) · Zbl 0171.38503
[2] Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and physicists, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Bd LXVII, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954. · Zbl 0055.11905
[3] Roland Bulirsch, Numerical calculation of elliptic integrals and elliptic functions, Numer. Math. 7 (1965), 78 – 90. · Zbl 0133.08702 · doi:10.1007/BF01397975
[4] Roland Bulirsch, Numerical calculation of elliptic integrals and elliptic functions. II, Numer. Math. 7 (1965), 353 – 354. · Zbl 0128.37204 · doi:10.1007/BF01436529
[5] R. Bulirsch, An extension of the Bartky-transformation to incomplete elliptic integrals of the third kind, Numer. Math. 13 (1969), 266 – 284. · Zbl 0175.46001 · doi:10.1007/BF02167558
[6] R. Bulirsch, Numerical calculation of elliptic integrals and elliptic functions. III, Numer. Math. 13 (1969), 305 – 315. · Zbl 0181.17502 · doi:10.1007/BF02165405
[7] B. C. Carlson, Normal elliptic integrals of the first and second kinds, Duke Math. J. 31 (1964), 405 – 419. · Zbl 0151.07501
[8] B. C. Carlson, On computing elliptic integrals and functions, J. Math. and Phys. 44 (1965), 36 – 51. · Zbl 0163.39601
[9] B. C. Carlson, Elliptic integrals of the first kind, SIAM J. Math. Anal. 8 (1977), no. 2, 231 – 242. · Zbl 0356.33002 · doi:10.1137/0508016
[10] B. C. Carlson, Short proofs of three theorems on elliptic integrals, SIAM J. Math. Anal. 9 (1978), no. 3, 524 – 528. · Zbl 0401.33001 · doi:10.1137/0509033
[11] B. C. Carlson, Computing elliptic integrals by duplication, Numer. Math. 33 (1979), no. 1, 1 – 16. · Zbl 0438.65029 · doi:10.1007/BF01396491
[12] Carlson, B.C., & Notis, E.M., Algorithm 577. Algorithms for Incomplete Elliptic Integrals, ACM Trans. Math. Software, 7, 398-403 (1981)
[13] B. C. Carlson and John L. Gustafson, Asymptotic expansion of the first elliptic integral, SIAM J. Math. Anal. 16 (1985), no. 5, 1072 – 1092. · Zbl 0593.33002 · doi:10.1137/0516080
[14] B. C. Carlson and J. L. Gustafson, Asymptotic approximations for symmetric elliptic integrals, SIAM J. Math. Anal. 25 (1994), no. 2, 288 – 303. · Zbl 0794.41021 · doi:10.1137/S0036141092228477
[15] W. J. Cody, Chebyshev approximations for the complete elliptic integrals \? and \?, Math. Comp. 19 (1965), 105 – 112. · Zbl 0137.33502
[16] W. J. Cody, Chebyshev polynomial expansions of complete elliptic integrals, Math. Comp. 19 (1965), 249 – 259. · Zbl 0134.33301
[17] W. J. Cody, Chebyshev approximations for the complete elliptic integrals \? and \?, Math. Comp. 19 (1965), 105 – 112. · Zbl 0137.33502
[18] A. R. DiDonato and A. V. Hershey, New formulas for computing incomplete elliptic integrals of the first and second kind, J. Assoc. Comput. Mach. 6 (1959), 515 – 526. · Zbl 0117.10804 · doi:10.1145/320998.321005
[19] Wyman G. Fair and Yudell L. Luke, Rational approximations to the incomplete elliptic integrals of the first and second kinds, Math. Comp. 21 (1967), 418 – 422. · Zbl 0171.37703
[20] Chelo Ferreira and José L. López, Symmetric standard elliptic integrals with two or three large parameters, Integral Transforms Spec. Funct. 17 (2006), no. 6, 433 – 442. · Zbl 1094.33015 · doi:10.1080/10652460600741009
[21] Frisch-Fay, R., Applications of Approximate Expressions for Complete Elliptic Integrals, Int. J. Mech. Sci., 5, 231-235 (1963)
[22] Toshio Fukushima, Fast computation of complete elliptic integrals and Jacobian elliptic functions, Celestial Mech. Dynam. Astronom. 105 (2009), no. 4, 305 – 328. · Zbl 1223.70005 · doi:10.1007/s10569-009-9228-z
[23] Toshio Fukushima, Fast computation of incomplete elliptic integral of first kind by half argument transformation, Numer. Math. 116 (2010), no. 4, 687 – 719. · Zbl 1201.65035 · doi:10.1007/s00211-010-0321-8
[24] Fukushima, T., Precise Computation of Acceleration due to Uniform Ring or Disk, Celest. Mech. Dyn. Astron., 108, 339-356 (2010b) · Zbl 1223.70040
[25] Fukushima, T., Precise and Fast Computation of General Complete Elliptic Integral of Second Kind, Math. Comp., in printing (2010c)
[26] Toshio Fukushima and Hideharu Ishizaki, Numerical computation of incomplete elliptic integrals of a general form, Celestial Mech. Dynam. Astronom. 59 (1994), no. 3, 237 – 251. · Zbl 0818.33013 · doi:10.1007/BF00692874
[27] D. J. Hofsommer and R. P. van de Riet, On the numerical calculation of elliptic integrals of the first and second kind and the elliptic functions of Jacobi, Numer. Math. 5 (1963), 291 – 302. · Zbl 0123.13204 · doi:10.1007/BF01385899
[28] D. Karp, A. Savenkova, and S. M. Sitnik, Series expansions for the third incomplete elliptic integral via partial fraction decompositions, J. Comput. Appl. Math. 207 (2007), no. 2, 331 – 337. · Zbl 1117.33013 · doi:10.1016/j.cam.2006.10.019
[29] D. Karp and S. M. Sitnik, Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity, J. Comput. Appl. Math. 205 (2007), no. 1, 186 – 206. · Zbl 1118.33010 · doi:10.1016/j.cam.2006.04.053
[30] José L. López, Asymptotic expansions of symmetric standard elliptic integrals, SIAM J. Math. Anal. 31 (2000), no. 4, 754 – 775. · Zbl 0994.33010 · doi:10.1137/S0036141099351176
[31] J. L. López, Uniform asymptotic expansions of symmetric elliptic integrals, Constr. Approx. 17 (2001), no. 4, 535 – 559. · Zbl 1086.41015 · doi:10.1007/s003650010038
[32] Yudell L. Luke, Approximations for elliptic integrals, Math. Comp. 22 (1968), 627 – 634. · Zbl 0165.17701
[33] Yudell L. Luke, Further approximations for elliptic integrals, Math. Comp. 24 (1970), 191 – 198. · Zbl 0195.45203
[34] Morris, A.H. Jr., NSWC Library of Mathematics Subroutines, Tech. Rep. NSWCDD/TR-92/425, 107-110. Naval Surface Warfare Center, Dahlgren (1993).
[35] Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark , NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). · Zbl 1198.00002
[36] William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling, Numerical recipes, Cambridge University Press, Cambridge, 1986. The art of scientific computing. William T. Vetterling, Saul A. Teukolsky, William H. Press, and Brian P. Flannery, Numerical recipes example book (Pascal), Cambridge University Press, Cambridge, 1985. William T. Vetterling, Saul A. Teukolsky, William H. Press, and Brian P. Flannery, Numerical recipes example book (FORTRAN), Cambridge University Press, Cambridge, 1985. · Zbl 0587.65003
[37] William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, Numerical recipes, 3rd ed., Cambridge University Press, Cambridge, 2007. The art of scientific computing. · Zbl 1132.65001
[38] H. Van de Vel, On the series expansion method for computing incomplete elliptic integrals of the first and second kinds, Math. Comp. 23 (1969), 61 – 69. · Zbl 0187.10201
[39] 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.