Integral representation of Kelvin functions and their derivatives with respect to the order

  • Alexander Apelblat
Original Papers


Integral representations of the Kelvin functions ber v x and bei v x and their derivatives with respect to the order are considered. Using the Laplace transform technique the derivatives are expressed in terms of finite integrals. The Kelvin functions bern+1/2x and bein+1/2x can be presented in a closed form.


Mathematical Method Closed Form Integral Representation Finite Integral 


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  1. [1]
    Lord Kelvin (Sir William Thomson),Ether, electricity and ponderable matter. Mathematical and Physical Papers,3, 484–515 (1890).Google Scholar
  2. [2]
    O. Heaviside,The induction of currents in cores.The Electrician 12, 583–586 (1884).Google Scholar
  3. [3]
    C. S. Whitehead,On the generalization of the functions berx, beix, kerx, keix. Quart. J. Pure Appl. Math.42, 316–342 (1911).Google Scholar
  4. [4]
    E. G. Richardson and E. Tyler,The transverse velocity gradient near the mouths of pipes in which alternating or continuous flow of air is established. Proc. Phys. Soc.42, 1–15 (1929).Google Scholar
  5. [5]
    E. Reissner,Stresses and small displacements of shallow spherical shells. II. J. Mathematics and Physics25, 279–300 (1946), Correction,27, 240 (1948).Google Scholar
  6. [6]
    J. R. Womersley,Method for calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiology127, 553–563 (1955).Google Scholar
  7. [7]
    G. N. Watson,A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge University Press, Cambridge 1958.Google Scholar
  8. [8]
    G. Petiau,La théorie des fonctions de Bessel. Centre National de la Recherche Scientifique, Paris 1955.Google Scholar
  9. [9]
    Y. Young and A. Kirk,Royal Society Mathematical Tables, vol. 10, Bessel Functions, Part IV, Kelvin Functions. Cambridge University Press, Cambridge 1963.Google Scholar
  10. [10]
    A. Abramowitz and I. E. Stegun,Handbook of Mathematical Functions. U.S. National Bureau of Standards, Washington, DC 1965.Google Scholar
  11. [11]
    N. W. McLachlan,Bessel Functions for Engineers, 2nd ed. The Clarendon Press, Oxford 1955.Google Scholar
  12. [12]
    A. Apelblat and N. Kravitsky,Integral representation of derivatives and integrals with respect to the order of the Bessel functions J v (t), I v (t), the Anger function J v (t) and the integral Bessel function Ji v (t). IMA J. Appl. Math.34, 187–210 (1985).Google Scholar
  13. [13]
    A. Apelblat,Derivatives and integrals with respect to the order of the Struve functions H v (t) and L v (t). J. Math. Anal. Appl.137, 17–36 (1989).Google Scholar
  14. [14]
    F. Oberhettinger and L. Badii,Tables of Laplace Transforms. Springer-Verlag, Berlin 1973.Google Scholar

Copyright information

© Birkhäuser Verlag 1991

Authors and Affiliations

  • Alexander Apelblat
    • 1
  1. 1.Department of Chemical EngineeringBen Gurion University of the NegevBeer ShevaIsrael

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