An Integral Transform Involving the Product Of Bessel Functions and Whittaker Function and Its Application
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An Integral Transform Involving the Product Of Bessel Functions and Whittaker Function and Its Application A. Belafhal1 · E. M. El Halba1 · T. Usman2 Accepted: 7 November 2020 © Springer Nature India Private Limited 2020
Abstract In this paper, we aim to establish a closed form for an integral transform involving the product of two Bessel functions of the first kind and the Whittaker function. This integral transform is evaluated in terms of the Lauricella’s triple hypergeometric function, which reduces to the generalized Kampé de Fériet function and is interesting to generate some new laser beams and to study the propagation of these waves through free space and atmospheric and maritime turbulent. Besides, some particular cases are derived by using the relations between the Whittaker function and other special functions and orthogonal polynomials. A novel beams family, called “ Lauricella beams ” is introduced and the propagation of Laguerre–Bessel–Gaussian beam through an ABCD optical system is studied. Keywords Integral transform · Bessel function · Whittaker function · Lauricella’s triple hypergeometric function · Generalized Kampé de Fériet function Mathematics Subject Classification 33B15 · 33C10 · 33C15
Introduction and Preliminaires During the last decades, important properties and applications of various integral transforms involving special functions were investigated (see [14,23,24,33]). Recently, many applications of doughnut beams with a dark spot at the center are investigated in optical tweezers [39], in optical trap of atoms [18,26,38] microscopic systems (see[1–7,16,32,40–43]) and in far-field fluorescence microscopy of biological samples [25]. These beams can be generated
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T. Usman [email protected] A. Belafhal [email protected] E. M. El Halba [email protected]
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LPNAMME, Laser Physics Group, Department of Physics, Faculty of Sciences, Chouaïb Doukkali University, P. B 20, 24000 El Jadida, Morocco
2
Department of Mathematics, School of Basic and Applied Sciences, Lingaya’s Vidyapeeth, Faridabad, Haryana 121002, India 0123456789().: V,-vol
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Int. J. Appl. Comput. Math
(2020) 6:177
by the use of a spiral phase plate [8,9,28] by an ABCD-treatment of the propagation of some fields which can be expressed as the integrand of an integral transform involving the product of two Bessel functions of the first kind and the Whittaker function. In this work, motivated by the recent result of Khan et al. (see [20–23]) and the applications published by Belafhal et al. [8] for the generation of generalized Humbert laser beams by conversion of circular beams by using a spiral phase plate and other investigations established last years [9,10], a closed form of this type of the considered integral transform is derived. We will derive this (3) closed form in terms of the Lauricella’s triple hypergeometric function FA (x, y, z), which reduces the generalized Kampé de Fériet [13]. For the present investigation, we recall the following definitions: The generalization of Gaus
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