Convergent expansions and bounds for the incomplete elliptic integral of the second kind near the logarithmic singularity
HTML articles powered by AMS MathViewer
- by Dmitrii Karp and Yi Zhang
- Math. Comp. 92 (2023), 2769-2794
- DOI: https://doi.org/10.1090/mcom/3874
- Published electronically: July 11, 2023
- HTML | PDF | Request permission
Abstract:
We find two series expansions for Legendre’s second incomplete elliptic integral $E(\lambda , k)$ in terms of recursively computed elementary functions. Both expansions converge at every point of the unit square in the $(\lambda , k)$ plane. Partial sums of the proposed expansions form a sequence of approximations to $E(\lambda ,k)$ which are asymptotic when $\lambda$ and/or $k$ tend to unity, including when both approach the logarithmic singularity $\lambda =k=1$ from any direction. Explicit two-sided error bounds are given at each approximation order. These bounds yield a sequence of increasingly precise asymptotically correct two-sided inequalities for $E(\lambda , k)$. For the reader’s convenience we further present explicit expressions for low-order approximations and numerical examples to illustrate their accuracy. Our derivations are based on series rearrangements, hypergeometric summation algorithms and extensive use of the properties of the generalized hypergeometric functions including some recent inequalities.References
- Bille C. Carlson’s Profile, https://dlmf.nist.gov/about/bio/BCCarlson
- Digital Library of Mathematical Functions, Chapter 19, Elliptic Integrals, https://dlmf.nist.gov/19
- S. I. Bezrodnykh, Analytic continuation of the Appell function $F_1$ and integration of the associated system of equations in the logarithmic case, Comput. Math. Math. Phys. 57 (2017), no. 4, 559–589. MR 3651114, DOI 10.1134/S0965542517040042
- 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. MR 277773, DOI 10.1007/978-3-642-65138-0
- B. C. Carlson, Some series and bounds for incomplete elliptic integrals, J. Math. and Phys. 40 (1961), 125–134. MR 130398, DOI 10.1002/sapm1961401125
- Billie Chandler Carlson, Special functions of applied mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. MR 590943
- B. C. Carlson, Computing elliptic integrals by duplication, Numer. Math. 33 (1979), no. 1, 1–16. MR 545738, DOI 10.1007/BF01396491
- B. C. Carlson, The hypergeometric function and the $R$-function near their branch points, Rend. Sem. Mat. Univ. Politec. Torino Special Issue (1985), 63–89. International conference on special functions: theory and computation (Turin, 1984). MR 850029
- B. C. Carlson and John L. Gustafson, Asymptotic expansion of the first elliptic integral, SIAM J. Math. Anal. 16 (1985), no. 5, 1072–1092. MR 800798, DOI 10.1137/0516080
- B. C. Carlson and J. L. Gustafson, Asymptotic approximations for symmetric elliptic integrals, SIAM J. Math. Anal. 25 (1994), no. 2, 288–303. MR 1266560, DOI 10.1137/S0036141092228477
- Frédéric Chyzak, An extension of Zeilberger’s fast algorithm to general holonomic functions, Discrete Math. 217 (2000), no. 1-3, 115–134 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR 1766263, DOI 10.1016/S0012-365X(99)00259-9
- 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. MR 239732, DOI 10.1090/S0025-5718-1969-0239732-8
- E. L. Kaplan, Auxiliary table for the incomplete elliptic integrals, J. Math. Physics 27 (1948), 11–36. MR 25247, DOI 10.1002/sapm194827111
- D. Karp, An approximation for zero-balanced Appell function $F_1$ near $(1,1)$, J. Math. Anal. Appl. 339 (2008), no. 2, 1332–1341. MR 2377090, DOI 10.1016/j.jmaa.2007.07.071
- 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. MR 2324834, DOI 10.1016/j.cam.2006.04.053
- D. Karp and S. M. Sitnik, Inequalities and monotonicity of ratios for generalized hypergeometric function, J. Approx. Theory 161 (2009), no. 1, 337–352. MR 2558159, DOI 10.1016/j.jat.2008.10.002
- C. Koutschan, HolonomicFunctions User’s Guide, RISC Report Series, pp. 1–93, 2010.
- José L. López, Asymptotic expansions of symmetric standard elliptic integrals, SIAM J. Math. Anal. 31 (2000), no. 4, 754–775. MR 1745143, DOI 10.1137/S0036141099351176
- J. L. López, Uniform asymptotic expansions of symmetric elliptic integrals, Constr. Approx. 17 (2001), no. 4, 535–559. MR 1845267, DOI 10.1007/s003650010038
- D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
- Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), 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). MR 2723248
- S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 (1997), no. 2, 278–301. MR 1600537, DOI 10.1112/S0025579300012602
- Doron Zeilberger, The method of creative telescoping, J. Symbolic Comput. 11 (1991), no. 3, 195–204. MR 1103727, DOI 10.1016/S0747-7171(08)80044-2
Bibliographic Information
- Dmitrii Karp
- Affiliation: Department of Mathematics, Holon Institute of Technology, Holon 5810201, Israel
- MR Author ID: 331068
- ORCID: 0000-0001-8206-3539
- Email: dimkrp@gmail.com
- Yi Zhang
- Affiliation: Department of Foundational Mathematics, School of Mathematics and Physics, Xi’an Jiaotong-Liverpool University, Suzhou 215123, People’s Republic of China
- Email: yi.zhang03@xjtlu.edu.cn
- Received by editor(s): August 12, 2022
- Received by editor(s) in revised form: February 20, 2023
- Published electronically: July 11, 2023
- Additional Notes: The second author was supported by the NSFC Young Scientist Fund No. 12101506, the Natural Science Foundation of the Jiangsu Higher Education Institutions of China No. 21KJB110032, and XJTLU Research Development Fund No. RDF-20-01-12.
The second author is the corresponding author. - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 2769-2794
- MSC (2020): Primary 33E05, 33F10, 33C20, 33C60
- DOI: https://doi.org/10.1090/mcom/3874