Acta Applicandae Mathematicae

, Volume 150, Issue 1, pp 141–177 | Cite as

Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions

  • Gergő NemesEmail author


In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise from an application of the method of steepest descents.


Asymptotic expansions Error bounds Remainder terms Bessel functions Hankel functions 

Mathematics Subject Classification

41A60 30E15 33C10 



The author’s research was supported by a research grant (GRANT11863412/ 70NANB15H221) from the National Institute of Standards and Technology. The author greatly appreciates the help of Dorottya Sziráki in improving the presentation of the paper. The author thanks the anonymous referees for their helpful comments and suggestions on the manuscript.


  1. 1.
    Airey, J.R.: “The converging factor” in asymptotic series and the calculation of Bessel, Laguerre and other functions. Philos. Mag. 24(7), 521–552 (1937) CrossRefzbMATHGoogle Scholar
  2. 2.
    Berry, M.V., Howls, C.J.: Hyperasymptotics for integrals with saddles. Proc. R. Soc. Lond. Ser. A 434(1892), 657–675 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bickley, W.G.: Formulæ relating to Bessel functions of moderate or large argument and order. Philos. Mag. 34(228), 37–49 (1943) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boyd, W.G.C.: Stieltjes transforms and the Stokes phenomenon. Proc. R. Soc. Lond. Ser. A 429(1876), 227–246 (1990) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boyd, W.G.C.: Error bounds for the method of steepest descents. Proc. R. Soc. Lond. Ser. A 440(1910), 493–518 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Burnett, D.: The remainders in the asymptotic expansions of certain Bessel functions. Math. Proc. Camb. Philos. Soc. 26(2), 145–151 (1930) CrossRefzbMATHGoogle Scholar
  7. 7.
    Dempsey, E., Benson, G.C.: Note on the asymptotic expansion of the modified Bessel function of the second kind. Math. Comput. 14(72), 362–365 (1960) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dingle, R.B.: Asymptotic expansions and converging factors. I. General theory and basic converging factors. Proc. R. Soc. Lond. Ser. A 244(1239), 456–475 (1958) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dingle, R.B.: Asymptotic expansions and converging factors IV. Confluent hypergeometric, parabolic cylinder, modified Bessel, and ordinary Bessel functions. Proc. R. Soc. Lond. Ser. A 249(1257), 270–283 (1959) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dingle, R.B.: Asymptotic Expansions: Their Derivation and Interpretation. Academic Press, New York (1973) zbMATHGoogle Scholar
  11. 11.
    Döring, B.: Über Fehlerschranken zu den Hankelschen asymptotischen Entwicklungen der Besselfunktionen für komplexes Argument und reellen Index. Z. Angew. Math. Mech. 42(1–2), 62–76 (1962) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goldstein, M., Thaler, R.M.: Bessel functions for large arguments. Math. Tables Other Aids Comput. 12(61), 18–26 (1958) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hamilton, W.R.: On fluctuating functions. Trans. R. Ir. Acad. 19, 264–321 (1843) Google Scholar
  14. 14.
    Hankel, H.: Die Cylinderfunctionen erster und zweiter Art. Math. Ann. 1(3), 467–501 (1869) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hansen, P.A.: Schriften der Sternwarte Seeberg: Ermittlung der Absoluten Störungen in Ellipsen von beliebiger Excentricität und Neigung. Gotha bei Carl Gläser (1843) Google Scholar
  16. 16.
    Heitman, Z., Bremer, J., Rokhlin, V., Vioreanu, B.: On the asymptotics of Bessel functions in the Fresnel regime. Appl. Comput. Harmon. Anal. 39(2), 347–356 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Howls, C.J.: Hyperasymptotics for integrals with finite endpoints. Proc. R. Soc. Lond. Ser. A 439(1906), 373–396 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jacobi, C.G.J.: Versuch einer Berechnung der grossen Ungleichheit des Saturns nach einer strengen Entwickelung. Astron. Nachr. 28(6), 81–94 (1849) CrossRefGoogle Scholar
  19. 19.
    Kirchhoff, G.: Ueber den inducirten Magnetismus eines unbegrenzten Cylinders von weichem Eisen. J. Reine Angew. Math. 48, 348–376 (1854) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Koshliakov, N.S.: Note on the reminders in the asymptotic expansions of Bessel functions. J. Lond. Math. Soc. s1-4(4), 297–299 (1929) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kummer, E.E.: De integralibus quibusdam definitis et seriebus infinitis. J. Reine Angew. Math. 17, 228–242 (1837) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lipschitz, R.: Ueber ein Integral der Differentialgleichung. J. Reine Angew. Math. 56, 189–196 (1859) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Meijer, C.S.: Asymptotische Entwicklungen von Besselschen, Hankelschen und verwandten Funktionen I–IV. In: Proc. Kon. Akad. Wet. Amsterdam, vol. 35, pp. 656–667, 852–866, 948–958, 1079–1090 (1932) Google Scholar
  24. 24.
    NIST Digital Library of Mathematical Functions., release 1.0.10 of 2015-08-07
  25. 25.
    Oberhettinger, F., Badii, L.: Tables of Laplace Transforms. Springer, Berlin (1973) CrossRefzbMATHGoogle Scholar
  26. 26.
    Olde Daalhuis, A.B., Olver, F.W.J.: Hyperasymptotic solutions of second-order linear differential equations I. Methods Appl. Anal. 2(2), 173–197 (1995) MathSciNetzbMATHGoogle Scholar
  27. 27.
    Olde Daalhuis, A.B., Olver, F.W.J.: Hyperasymptotic solutions of second-order linear differential equations II. Methods Appl. Anal. 2(2), 198–211 (1995) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Olver, F.W.J.: Error bounds for asymptotic expansions, with an application to cylinder functions of large argument. In: Wilcox, C.H. (ed.) Asymptotic Solutions of Differential Equations and Their Applications, pp. 163–183. Wiley, New York (1964) Google Scholar
  29. 29.
    Olver, F.W.J.: Asymptotic approximations and error bounds. SIAM Rev. 22(2), 188–203 (1980) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Olver, F.W.J.: On Stokes’ phenomenon and converging factors. In: Wong, R. (ed.) Asymptotic and Computational Analysis. Lecture Notes in Pure and Appl. Math., vol. 124, pp. 329–355. Marcel Dekker, New York (1990) Google Scholar
  31. 31.
    Olver, F.W.J.: Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral. SIAM J. Math. Anal. 22(5), 1460–1474 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Olver, F.W.J.: Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms. SIAM J. Math. Anal. 22(5), 1475–1489 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Olver, F.W.J.: Asymptotics and Special Functions, AKP Class. AK Peters, Wellesley (1997). Reprint of the original (1974) edition Google Scholar
  34. 34.
    Paris, R.B.: Exponentially small expansions of the confluent hypergeometric functions. Appl. Math. Sci. 7(133), 6601–6609 (2013) MathSciNetGoogle Scholar
  35. 35.
    Poisson, S.-D.: Suite du mémoire sur la distribution de la chaleur dans les corps solides. J. Éc. Polytech. 12(19), 249–403 (1823) Google Scholar
  36. 36.
    Schläfli, L.: Sull’uso delle linee lungo le quali il valore assoluto di una funzione è constante. Ann. Mat. 2(6), 1–20 (1875) Google Scholar
  37. 37.
    Stieltjes, T.J.: Recherches sur quelques séries semi-convergentes. Ann. Sci. Éc. Norm. Supér. 3(3), 201–258 (1886) CrossRefzbMATHGoogle Scholar
  38. 38.
    Tao, T.: An Introduction to Measure Theory. Grad. Studies in Math., vol. 126. Am. Math. Soc., Providence (2011) zbMATHGoogle Scholar
  39. 39.
    Temme, N.M.: Asymptotic Methods for Integrals. Ser. Analysis, vol. 6. World Scientific, Singapore (2015) zbMATHGoogle Scholar
  40. 40.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge Math. Lib. Cambridge University Press, Cambridge (1995). Reprint of the second (1944) edition zbMATHGoogle Scholar
  41. 41.
    Weber, H.: Zur Theorie der Bessel’schen Functionen. Math. Ann. 37(3), 404–416 (1890) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.School of MathematicsThe University of EdinburghEdinburghUK

Personalised recommendations