Geodesic
An integral transform and its application in the propagation of Lorentz-Gaussian beams
Communications in Mathematics, Tome 29 (2021) no. 3, pp. 483-491

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The aim of the present note is to derive an integral transform $$I=\int _{0}^{\infty } x^{s+1} e^{-\beta x^{2}+\gamma x} M_{k, \nu }\left (2 \zeta x^{2}\right )J_{\mu }(\chi x) dx,$$ involving the product of the Whittaker function $M_{k, \nu }$ and the Bessel function of the first kind $J_{\mu }$ of order $\mu $. As a by-product, we also derive certain new integral transforms as particular cases for some special values of the parameters $k$ and $\nu $ of the Whittaker function. Eventually, we show the application of the integral in the propagation of hollow higher-order circular Lorentz-cosh-Gaussian beams through an ABCD optical system (see, for details \cite {Xu2019}, \cite {Collins1970}).
Classification : 33B15, 33C10, 33C15
Keywords: Integral transform; Bessel function; Whittaker function; Confluent hypergeometric function; Lorentz-Gaussian beams. 
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     title = {An integral transform and its application in the propagation of {Lorentz-Gaussian} beams},
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Belafhal, A.; Halba, E.M. El; Usman, T. An integral transform and its application in the propagation of Lorentz-Gaussian beams. Communications in Mathematics, Tome 29 (2021) no. 3, pp. 483-491. http://geodesic.mathdoc.fr/item/COMIM_2021__29_3_a11/