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.
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Agrest, M.M., Maksimov, M.S.: Theory of Incomplete Cylindrical Functions and their Applications. Springer, Berlin (1971)
Apelblat, A.: Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42(5), 708–714 (1991)
Baricz, Á., Jankov, D., Pogány, T.K.: Neumann series of Bessel functions. Integral Transforms Spec. Funct. 23(7), 529–538 (2012)
Born, M., Wolf, E.: Principles of Optics. Pergamon Press, Oxford (1965)
Bray, W.O., Stanojević, V.B.: On the integrability of complex trigonometric series. Proc. Am. Math. Soc. 93(1), 51–58 (1985)
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)
Brualla, L., Martin, P.: Analytic approximations to Kelvin functions with applications to electromagnetics. J. Phys. A 34(43), 9153 (2001)
De Micheli, E., Viano, G.A.: Holomorphic extension associated with Fourier–Legendre expansions. J. Geom. Anal. 12(3), 355–374 (2002)
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)
Dras̆c̆ić, B., Pogány, T.K.: On integral representation of Bessel function of the first kind. J. Math. Anal. Appl. 308(2), 775–780 (2005)
Flesch, R.J., Trullinger, S.E.: Green’s functions for nonlinear Klein–Gordon kink perturbation theory. J. Math. Phys. 28(7), 1619–1631 (1987)
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)
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)
Jardim, R.F., Laks, B.: Kelvin functions for determination of magnetic susceptibility in nonmagnetic metals. J. Appl. Phys. 65(12), 4505 (1989)
Korenev, B.G.: Bessel Functions and their Applications. Chapman & Hall/CRC, Boca Raton (2002)
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)
Luke, Y.L.: Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13, 261–271 (1959)
Luke, Y.L.: The Special Functions and their Approximations, vol. 2. Academic Press, New York (1969)
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)
Paris, R.B.: High-precision evaluation of the Bessel functions via Hadamard series. J. Comput. Appl. Math. 224(1), 84–100 (2009)
Pogány, T.K., Süli, E.: Integral representation of Neumann series of Bessel functions. Proc. Am. Math. Soc. 137(7), 2363–2368 (2009)
Rice, S.O.: Mathematical analysis of random noise. III. Bell Syst. Tech. J. 24(1), 46–156 (1945)
Szegö, G.: Orthogonal Polynomials. American Mathematical Society, Providence (1975)
Tricomi, F.G.: Asymptotische Eigenschaften der unvollständigen Gammafunktion. Math. Z. 53(2), 136–148 (1950)
Tricomi, F.G.: Sulla funzione gamma incompleta. Ann. Math. Pura Appl. 31(1), 263–279 (1950)
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)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)
Wilkins Jr., J.E.: Neumann series of Bessel functions. Trans. Am. Math. Soc. 64(2), 359–385 (1948)
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This work has been partially funded by C.N.R. - Italy, Project MD.P01.004.001.
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De Micheli, E. Integral Representation for Bessel’s Functions of the First Kind and Neumann Series. Results Math 73, 61 (2018). https://doi.org/10.1007/s00025-018-0826-5
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DOI: https://doi.org/10.1007/s00025-018-0826-5