Integral representation of Kelvin functions and their derivatives with respect to the order. (English) Zbl 0741.33002

Apelblat, Alexander

Z. Angew. Math. Phys. 42, No. 5, 708-714 (1991).

Integral representations of the Kelvin functions \(\hbox{ber}_ \nu x\) and \(\hbox{bei}_ \nu x\) 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 \(\hbox{ber}_{n+1/2}x\) and \(\hbox{bei}_{n+1/2}x\) can be presented in a closed form.

Reviewer: H.Köditz (Hannover)

MSC:

33C10

Bessel and Airy functions, cylinder functions, \({}_0F_1\)

Keywords:

Kelvin functions

References:

[1]	Lord Kelvin (Sir William Thomson),Ether, electricity and ponderable matter. Mathematical and Physical Papers,3, 484-515 (1890).
[2]	O. Heaviside,The induction of currents in cores.The Electrician 12, 583-586 (1884).
[3]	C. S. Whitehead,On the generalization of the functions berx, beix, kerx, keix. Quart. J. Pure Appl. Math.42, 316-342 (1911).
[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). · JFM 55.1147.04
[5]	E. Reissner,Stresses and small displacements of shallow spherical shells. II. J. Mathematics and Physics25, 279-300 (1946), Correction,27, 240 (1948). · Zbl 0060.42407
[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).
[7]	G. N. Watson,A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge University Press, Cambridge 1958. · Zbl 0083.20702
[8]	G. Petiau,La théorie des fonctions de Bessel. Centre National de la Recherche Scientifique, Paris 1955. · Zbl 0067.04704
[9]	Y. Young and A. Kirk,Royal Society Mathematical Tables, vol. 10, Bessel Functions, Part IV, Kelvin Functions. Cambridge University Press, Cambridge 1963. · Zbl 0147.36206
[10]	A. Abramowitz and I. E. Stegun,Handbook of Mathematical Functions. U.S. National Bureau of Standards, Washington, DC 1965.
[11]	N. W. McLachlan,Bessel Functions for Engineers, 2nd ed. The Clarendon Press, Oxford 1955. · JFM 61.1177.05
[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). · Zbl 0583.33006
[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). · Zbl 0669.33009
[14]	F. Oberhettinger and L. Badii,Tables of Laplace Transforms. Springer-Verlag, Berlin 1973. · Zbl 0285.65079