The main aim of this research note is to establish two different types of integral representation formulae for the Kapteyn series of the first kind. The first one is a double definite integral expression, while the second type includes indefinite integral representation formula.
REFERENCES
1.
Bessel
, F. W.
, “Analytische Auflösung der Keplerschen Aufgabe
,” Abhandlungen der Mathematischen Klasse der Königlich Preußischen Akademie der Wissenschaften zu Berlin. Aus der Jahren
1816–1817
, 49
–55
(1819
).2.
Chessin
, A. S.
, “Sur l'équation de Bessel avec second membre
,” Compt. Rend.
135
, 678
–679
(1902
).3.
Chessin
, A. S.
, “Sur une classe d'équations différentielles réductibles a l'équation de Bessel
,” Compt. Rend.
136
, 1124
(1903
).4.
Citrin
, D. S.
, “Optical analogue for phase-sensitive measurements in quantum-transport experiments
,” Phys. Rev. B
60
(8
), 5659
(1999
).5.
Colwell
, P.
, “Bessel functions and Kepler's equation
,” Am. Math. Monthly
99
(1
), 45
(1992
).6.
Dominici
, D.
, “A new Kapteyn series
,” Integral Transforms Spec. Funct.
18
, 409
(2007
).7.
Dominici
, D.
, “An application of Kapteyn series to a problem from queueing theory
,” Proc. Appl. Math. Mech.
7
(1
), 2050005
(2007
).8.
Dominici
, D.
, “On Taylor series and Kapteyn series of the first and second type
,” e-print arXiv:1012.3700v1.9.
Eisinberg
, A.
, Fedele
, G.
, Ferrise
, A.
, and Frascino
, D.
, “On an integral representation of a class of Kapteyn (Fourier–Bessel) series: Kepler's equation, radiation problems and Meissel's expansion
,” Appl. Math. Lett.
23
(11
), 1331
(2010
).10.
Gradshteyn
, I. S.
and Ryzhik
, I. M.
, Table of Integrals, Series, and Products
, 6th ed. (Academic
, San Diego, CA
, 2000
).11.
Jankov
, D.
, Pogány
, T. K.
, and Süli
, E.
, “On the coefficients of Neumann series of Bessel functions
,” J. Math. Anal. Appl.
(2011
).12.
Kapteyn
, W.
, “Recherches sur les functions de Fourier–Bessel
,” Ann. Sci. de l'École Norm. Sup.
10
, 91
(1893
).13.
Kapteyn
, W.
, “On an expansion of an arbitrary function in a series of Bessel functions
,” Messenger Math.
35
, 122
(1906
).14.
Karamata
, J.
, Theory and Application of the Stieltjes Integral
, Srpska Akademija Nauka, Posebna izdanja CLIV, Matematički institut
, Knjiga I, Beograd
, 1949
(in Serbian)15.
Lagrange
, J. L.
, “Sur le probléme de Kepler
,” Mèm. de l'Acad. des Sci. Berlin
XXIV
, 204
(1771
).16.
Landau
, L.
, “Monotonicity and bounds on Bessel functions
,” in Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory
, Berkeley, California
, 2000
, pp. 147
–154
;Landau
, L.
, Electron. J. Differ. Equations
4
(2002
), Southwest Texas State University, San Marcos, TX.17.
Lerche
, I.
, Schlickeiser
, R.
, and Tautz
, R. C.
, “Comment on a new derivation of the plasma susceptibility tensor for a hot magnetized plasma without infinite sums of products of Bessel functions
,” Phys. Plasmas
15
(2
), 024701
(2008
).18.
Lerche
, I.
and Tautz
, R. C.
, “A note on summation of Kapteyn series in astrophysical problems
,” Astrophys. J.
665
, 1288
(2007
).19.
Lerche
, I.
and Tautz
, R. C.
, “Kapteyn series arising in radiation problems
,” J. Phys. A
41
(3
), 035202
(2008
).20.
Lerche
, I.
, Tautz
, R. C.
, and Citrin
, D. S.
, “Terahertz-sideband spectra involving Kapteyn series
,” J. Phys. A
42
(36
), 365206
(2009
).21.
Nielsen
, N.
, “Recherches sur les séries de fonctions cylindriques dues á C. Neumann et W. Kapteyn
,” Ann. Sci. de l'École Norm. Sup.
18
, 39
(1901
).22.
Pogány
, T. K.
and Süli
, E.
, “Integral representation for Neumann series of Bessel functions
,” Proc. Am. Math. Soc.
137
(7
), 2363
(2009
).23.
Schott
, G. A.
, Electromagnetic Radiation and the Mechanical Reactions Arising From It, Being an Adams Prize Essay in the University of Cambridge
(Cambridge University Press
, Cambridge
, 1912
).24.
Shalchi
, A.
andSchlickeiser
, R.
, “Cosmic ray transport in anisotropic magnetohydrodynamic turbulence III. Mixed magnetosonic and Alfvènic turbulence
,” Astron. Astrophys.
420
(3
), 799
(2004
).25.
Siemon
, P.
, “Über die Integrale einer nicht homogenen Differentialgleichung zweiter Ordnung
,” in Programm der Luisienschuhle. Sechster Abschnitt. Differential- und Integralrechnung.
Capitel 5. Gewöhnliche Differentialgleichungen
, Berlin
, 1890
.26.
Tautz
, R. C.
and Lerche
, I.
, “A review of procedures for summing Kapteyn series in mathematical physics
,” Adv. Math. Phys.
2009
, 425164
(2009
).27.
Thomson
, J. J.
, “The magnetic properties of systems of corpuscles describing circular orbits
,” Philos. Mag.
6
, 673
(1903
).28.
Watson
, G. N.
, A Treatise on the Theory of Bessel Functions
(Cambridge University Press
, Cambridge
, 1922
).© 2011 American Institute of Physics.
2011
American Institute of Physics
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