| Title:
|
An integral transform and its application in the propagation of Lorentz-Gaussian beams (English) |
| Author:
|
Belafhal, A. |
| Author:
|
Halba, E.M. El |
| Author:
|
Usman, T. |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
29 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
483-491 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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}). (English) |
| Keyword:
|
Integral transform |
| Keyword:
|
Bessel function |
| Keyword:
|
Whittaker function |
| Keyword:
|
Confluent hypergeometric function |
| Keyword:
|
Lorentz-Gaussian beams. |
| MSC:
|
33B15 |
| MSC:
|
33C10 |
| MSC:
|
33C15 |
| idZBL:
|
Zbl 07484382 |
| idMR:
|
MR4355423 |
| . |
| Date available:
|
2022-01-10T10:10:14Z |
| Last updated:
|
2022-04-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149331 |
| . |
| Reference:
|
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| Reference:
|
[2] Chen, R., An, C.: On the evaluation of infinite integrals involving Bessel functions.App. Math. Comput., 235, 2014, 212-220, MR 3194597, 10.1016/j.amc.2014.03.016 |
| Reference:
|
[3] Collins, S.A.: Lens-system diffraction integral written in terms of matrix optics.J. Opt. Soc. Am., 60, 9, 1970, 1168-1177, 10.1364/JOSA.60.001168 |
| Reference:
|
[4] Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products (5th edition).1994, Academic Press Inc., Boston, |
| Reference:
|
[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 3413773 |
| Reference:
|
[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, |
| Reference:
|
[7] Rainville, E.D.: Intermediate Differential Equations.1964, Macmillan, |
| Reference:
|
[8] Rainville, E.D.: Special Functions.1960, Macmillan Company, New York. Reprinted by Chelsea Publishing Company, Bronx, New York (1971), |
| Reference:
|
[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 0535.33001 |
| Reference:
|
[10] Watson, G.N.: A Treatise on the Theory of Bessel Functions (second edition).1944, Cambridge University Press, Cambridge, |
| Reference:
|
[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, 10.1090/S0002-9904-1903-01077-5 |
| Reference:
|
[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, |
| Reference:
|
[13] Xu, Y., Zhou, G.: Circular Lorentz-Gauss beams.J. Opt. Soc. Am. A., 36, 2, 2019, 179-185, 10.1364/JOSAA.36.000179 |
| . |
