Integral Representation for Bessel’s Functions of the First Kind and Neumann Series

Article
  • 18 Downloads

Abstract

A Fourier-type integral representation for Bessel’s functions of the first kind and complex order is obtained by using the Gegenbauer extension of Poisson’s integral representation for the Bessel function along with a suitable trigonometric integral representation of Gegenbauer’s polynomials. By using this representation, expansions in series of Bessel’s functions of various functions which are related to the incomplete gamma function can be obtained in a unified way. Neumann series are then considered and a new closed-form integral representation for this class of series is given. The density function of this representation is the generating function of the sequence of coefficients of the Neumann series on the unit circle. Examples of new closed-form integral representations of special functions are thus presented.

Keywords

Bessel function of the first kind Neumann series integral representation special function 

Mathematics Subject Classification

Primary 33C10 40C10 Secondary 33B20 

References

  1. 1.
    Agrest, M.M., Maksimov, M.S.: Theory of Incomplete Cylindrical Functions and their Applications. Springer, Berlin (1971)CrossRefMATHGoogle Scholar
  2. 2.
    Apelblat, A.: Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42(5), 708–714 (1991)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Baricz, Á., Jankov, D., Pogány, T.K.: Neumann series of Bessel functions. Integral Transforms Spec. Funct. 23(7), 529–538 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Born, M., Wolf, E.: Principles of Optics. Pergamon Press, Oxford (1965)Google Scholar
  5. 5.
    Bray, W.O., Stanojević, V.B.: On the integrability of complex trigonometric series. Proc. Am. Math. Soc. 93(1), 51–58 (1985)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bros, J., Viano, G.A.: Connection between the harmonic analysis on the sphere and the harmonic analysis on the one-sheeted hyperboloid: an analytic continuation viewpoint III. Forum Math. 9, 165–191 (1995)MathSciNetMATHGoogle Scholar
  7. 7.
    Brualla, L., Martin, P.: Analytic approximations to Kelvin functions with applications to electromagnetics. J. Phys. A 34(43), 9153 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    De Micheli, E., Viano, G.A.: Holomorphic extension associated with Fourier–Legendre expansions. J. Geom. Anal. 12(3), 355–374 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    De Micheli, E., Viano, G.A.: The expansion in Gegenbauer polynomials: a simple method for the fast computation of the Gegenbauer coefficients. J. Comput. Phys. 239, 112–122 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dras̆c̆ić, B., Pogány, T.K.: On integral representation of Bessel function of the first kind. J. Math. Anal. Appl. 308(2), 775–780 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Flesch, R.J., Trullinger, S.E.: Green’s functions for nonlinear Klein–Gordon kink perturbation theory. J. Math. Phys. 28(7), 1619–1631 (1987)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gautschi, W.: The incomplete gamma functions since Tricomi. In: Tricomi’s Ideas and Contemporary Applied Mathematics. Atti dei Convegni Lincei 147, pp. 203–237. Accademia Nazionale dei Lincei, Roma (1998)Google Scholar
  13. 13.
    Jankov, D., Pogány, T.K., Süli, E.: On the coefficients of Neumann series of Bessel functions. J. Math. Anal. Appl. 380(2), 628–631 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jardim, R.F., Laks, B.: Kelvin functions for determination of magnetic susceptibility in nonmagnetic metals. J. Appl. Phys. 65(12), 4505 (1989)CrossRefGoogle Scholar
  15. 15.
    Korenev, B.G.: Bessel Functions and their Applications. Chapman & Hall/CRC, Boca Raton (2002)MATHGoogle Scholar
  16. 16.
    Kravchenko, V.V., Torba, S.M., Castillo-Pérez, R.: A Neumann series of Bessel functions representation for solutions of perturbed Bessel equations. Appl. Anal. 97(5), 677–704 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Luke, Y.L.: Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13, 261–271 (1959)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Luke, Y.L.: The Special Functions and their Approximations, vol. 2. Academic Press, New York (1969)MATHGoogle Scholar
  19. 19.
    NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.15. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V. (eds)
  20. 20.
    Paris, R.B.: High-precision evaluation of the Bessel functions via Hadamard series. J. Comput. Appl. Math. 224(1), 84–100 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Pogány, T.K., Süli, E.: Integral representation of Neumann series of Bessel functions. Proc. Am. Math. Soc. 137(7), 2363–2368 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Rice, S.O.: Mathematical analysis of random noise. III. Bell Syst. Tech. J. 24(1), 46–156 (1945)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Szegö, G.: Orthogonal Polynomials. American Mathematical Society, Providence (1975)MATHGoogle Scholar
  24. 24.
    Tricomi, F.G.: Asymptotische Eigenschaften der unvollständigen Gammafunktion. Math. Z. 53(2), 136–148 (1950)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Tricomi, F.G.: Sulla funzione gamma incompleta. Ann. Math. Pura Appl. 31(1), 263–279 (1950)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Veling, E.J.M.: The generalized incomplete gamma function as sum over modified Bessel functions of the first kind. J. Comput. Appl. Math. 235(14), 4107–4116 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)MATHGoogle Scholar
  28. 28.
    Wilkins Jr., J.E.: Neumann series of Bessel functions. Trans. Am. Math. Soc. 64(2), 359–385 (1948)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IBF - Consiglio Nazionale delle RicercheGenovaItaly

Personalised recommendations