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 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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}).
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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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/

[1] Andrews, G.E., Askey, R., Roy, R.: Special Functions. 1999, Encyclopedia of Mathematics and its Applications 71. Cambridge University Press, Cambridge,

[2] Chen, R., An, C.: On the evaluation of infinite integrals involving Bessel functions. App. Math. Comput., 235, 2014, 212-220, | DOI | MR

[3] Collins, S.A.: Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am., 60, 9, 1970, 1168-1177, | DOI

[4] Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products (5th edition). 1994, Academic Press Inc., Boston,

[5] Khan, N.U., Kashmin, T.: On infinite series of three variables involving Whittaker and Bessel functions. Palest. J. Math., 5, 1, 2015, 185-190, | MR

[6] Khan, N.U., Usman, T., Ghayasuddin, M.: A note on integral transforms associated with Humbert's confluent hypergeometric function. Electron. J. Math. Anal. Appl., 4, 2, 2016, 259-265,

[7] Rainville, E.D.: Intermediate Differential Equations. 1964, Macmillan,

[8] Rainville, E.D.: Special Functions. 1960, Macmillan Company, New York. Reprinted by Chelsea Publishing Company, Bronx, New York (1971),

[9] Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. 1984, Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press, New York, | Zbl

[10] Watson, G.N.: A Treatise on the Theory of Bessel Functions (second edition). 1944, Cambridge University Press, Cambridge,

[11] Whittaker, E.T.: An expression of certain known functions as generalized hypergeometric functions. Bull. Amer. Math. Soc., 10, 3, 1903, 125\IL2\textendash 134, | DOI

[12] Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis (reprint of the fourth (1927) edition). 1996, Cambridge Mathematical Library, Cambridge University Press, Cambridge,

[13] Xu, Y., Zhou, G.: Circular Lorentz-Gauss beams. J. Opt. Soc. Am. A., 36, 2, 2019, 179-185, | DOI