An integral transform and its application in the propagation of Lorentz-Gaussian beams

A. Belafhal; E.M. El Halba; T. Usman

Communications in Mathematics (2021)

  • Volume: 29, Issue: 3, page 483-491
  • ISSN: 1804-1388

Abstract

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The aim of the present note is to derive an integral transform I = ∫ 0 ∞ x s + 1 e - β x 2 + γ x M k , ν 2 ζ x 2 J μ ( χ x ) d x , involving the product of the Whittaker function M k , ν and the Bessel function of the first kind J μ of order μ . As a by-product, we also derive certain new integral transforms as particular cases for some special values of the parameters k and ν 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 [Xu2019], [Collins1970]).

How to cite

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Belafhal, A., Halba, E.M. El, and Usman, T.. "An integral transform and its application in the propagation of Lorentz-Gaussian beams." Communications in Mathematics 29.3 (2021): 483-491. <http://eudml.org/doc/297517>.

@article{Belafhal2021,
abstract = {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 [Xu2019], [Collins1970]).},
author = {Belafhal, A., Halba, E.M. El, Usman, T.},
journal = {Communications in Mathematics},
keywords = {Integral transform; Bessel function; Whittaker function; Confluent hypergeometric function; Lorentz-Gaussian beams},
language = {eng},
number = {3},
pages = {483-491},
publisher = {University of Ostrava},
title = {An integral transform and its application in the propagation of Lorentz-Gaussian beams},
url = {http://eudml.org/doc/297517},
volume = {29},
year = {2021},
}

TY - JOUR
AU - Belafhal, A.
AU - Halba, E.M. El
AU - Usman, T.
TI - An integral transform and its application in the propagation of Lorentz-Gaussian beams
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
VL - 29
IS - 3
SP - 483
EP - 491
AB - 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 [Xu2019], [Collins1970]).
LA - eng
KW - Integral transform; Bessel function; Whittaker function; Confluent hypergeometric function; Lorentz-Gaussian beams
UR - http://eudml.org/doc/297517
ER -

References

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  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, (1984) Zbl0535.33001
  10. Watson, G.N., A Treatise on the Theory of Bessel Functions (second edition), 1944, Cambridge University Press, Cambridge, (1944) 
  11. Whittaker, E.T., 10.1090/S0002-9904-1903-01077-5, Bull. Amer. Math. Soc., 10, 3, 1903, 1252–134, (1903) DOI10.1090/S0002-9904-1903-01077-5
  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, (1996) 
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