×

zbMATH — the first resource for mathematics

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

MSC:
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
PDF BibTeX XML Cite
Full Text: DOI
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 · doi:10.1088/0959-5309/42/1/302
[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 · doi:10.1093/imamat/34.2.187
[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 · doi:10.1016/0022-247X(89)90270-9
[14] F. Oberhettinger and L. Badii,Tables of Laplace Transforms. Springer-Verlag, Berlin 1973. · Zbl 0285.65079
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.