Skip to main content
SpringerLink
Log in

An Integral Transform Involving the Product Of Bessel Functions and Whittaker Function and Its Application

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdalla, M.S., Abdel-Aty, M., Obada, A.-S.F.: Degree of entanglement for anisotropic coupled oscillators interacting with a single atom. J. Opt. B Quantum Semiclassical Opt. 4(6), 396–401 (2002)

    Google Scholar 

  2. Abdel-Aty, M.: Quantum field entropy and entanglement of a three-level atom two-mode system with an arbitrary nonlinear medium. J. Mod. Opt. 50(2), 161–177 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Abdel-Aty, M.: General formalism of interaction of a two-level atom with cavity field in arbitrary forms of nonlinearities. Physica A 313(3–4), 471–487 (2002)

    MATH  Google Scholar 

  4. Abdalla, M.S., Hassan, S.S., Abdel-Aty, M.: Entropic uncertainty in the Jaynes–Cummings model in presence of a second harmonic generation. Opt. Commun. 244(1–6), 431–443 (2005)

    Google Scholar 

  5. Abdel-Aty, M., Abdalla, M.S., Obada, A.-S.F.: Uncertainty relation and information entropy of a time-dependent bimodal two-level system. J. Phys. B: At. Mol. Opt. Phys. 35(23), 4773–4786 (2002)

    Google Scholar 

  6. Abdel-Aty, M., Abdel-Khalek, S., Obada, A.-S.F.: Pancharatnam phase of two-mode optical fields with Kerr nonlinearity. Opt. Rev. 7(6), 499–504 (2000)

    Google Scholar 

  7. Abdel-Aty, M., Abdalla, M.S., Obada, A.-S.F.: Entropy and phase properties of isotropic coupled oscillators interacting with a single atom: one- and two-photon processes. J. Opt. B Quantum Semiclassical Opt. 4(3), S133–S141 (2002)

    Google Scholar 

  8. Belafhal, A., Saad, F.: Conversion of circular beams by a spiral phase plate: generation of generalized Humbert beams. Optik 138, 516–528 (2017)

    Google Scholar 

  9. Belafhal, A., Nebdi, H.: Generation and propagation of novel donut beams by a spiral phase plate: Humbert beams. Opt. Quant. Electron. 46, 201–208 (2014)

    Google Scholar 

  10. Belafhal, A., Hennani, S.: A note on some integrals used in laser field involving the product of Bessel functions. Phys. Chem. News 61, 59–62 (2011)

    Google Scholar 

  11. Belafhal, A., Dalil-Essakali, L.: Collins formula and propagation of Bessel-modulated Gaussian light beams through an ABCD optical system. Opt. Commun. 177, 181–188 (2000)

    Google Scholar 

  12. Caron, C.F.R., Potvliege, R.M.: Bessel-modulated Gaussian beams with quadratic radial dependence. Opt. Commun. 164, 83–93 (1999)

    Google Scholar 

  13. Choi, J., Rathie, A.K.: Relations between Lauricella’s triple hypergeometric function \({F}_{A}^{(3)}(x, y, z)\) and Exton’s function \({X}_8\). Adv. Differ. Equ. 34, 1–7 (2013)

    MATH  Google Scholar 

  14. Choi, J., Agarwal, P., Mathur, S., Purohit, S.D.: Certain new integral formulas involving the generalized Bessel functions. Bull. Korean Math. Soc 51, 995–1003 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Collins, S.A.: Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 60, 1168–1177 (1970)

    Google Scholar 

  16. Farouk, A., Batle, J., Elhoseny, M., Naseri, M., Lone, M., Fedorov, A., Alkhambashi, M., Majid, A., Ahmed, S.H., Abdel-Aty, M.: Robust general N user authentication scheme in a centralized quantum communication network via generalized GHZ states. Front. Phys. 13(2), 130306 (2018)

    Google Scholar 

  17. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 5th edn. Academic Press, New York (1994)

    MATH  Google Scholar 

  18. Kaplan, A., Friedman, N., Davidson, N.: Optimized single-beam dark optical trap. J. Opt. Soc. Am. B 19, 1233–1238 (2002)

    Google Scholar 

  19. Karlsson, P.W.: Some reducible generalized Kampé de Fériet functions. J. Math. Anal. Appl. 96, 546–550 (1983)

    MathSciNet  MATH  Google Scholar 

  20. Khan, N.U., Usman, T., Aman, M.: Some proprties concerning the analysis of generalized Wright function. J. Comput. Appl. Math. 376, 112840 (2020)

    MathSciNet  MATH  Google Scholar 

  21. Khan, N.U., Usman, T., Ghayasuddin, M.: A new class of unified integral formulas associated with whittaker functions. NTMSCI 4(1), 160–167 (2016)

    MathSciNet  Google Scholar 

  22. Khan, N.U., Nadeem, R., Usman, T., Khan, A.H.: Evaluation of integral formulas associated with the product of generalized Bessel function with orthogonal polynomials. Honam Math. J. 41(1), 135–152 (2019)

    MathSciNet  MATH  Google Scholar 

  23. 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, 259–265 (2016)

    MATH  Google Scholar 

  24. Nisar, K.S., Suthar, D.L., Purohit, S.D., Aldhaifallah, M.: Some unified integrals associated with the generalized Struve function. Proc. Jangjeon Math. Soc. 20, 261–267 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Klar, T.A., Hell, S.W.: Subdiffraction resolution in far-field fluorescence microscopy. Opt. Lett. 24, 954–956 (1999)

    Google Scholar 

  26. Kuga, T., Torii, Y., Shiokawa, N., Hirano, T.: Novel optical trap of atoms with a doughnut beam. Phys. Rev. Lett. 78, 4713–4716 (1997)

    Google Scholar 

  27. Mandel, L., Wolf, E.: Optical Coherence and Quantum Optics, 2nd edn. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  28. Mawardi, A., Hild, S., Widera, A., Meschede, D.: ABCD-treatment of a propagating doughnut beam generated by a spiral phase plate. Opt. Express 19, 21205–21210 (2011)

    Google Scholar 

  29. Mei, Z., Zhao, D.: Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams. Opt. Express 15, 11942–11951 (2007)

    Google Scholar 

  30. Rainville, E.D.: Special Functions. The MacMillan Comp., New York, NY (1960)

    MATH  Google Scholar 

  31. Saad, F., Hricha, Z., Khouilid, M., Belafhal, A.: Propagation of Laguerre–Bessel–Gaussian beams. Optik 149, 412–416 (2017)

    Google Scholar 

  32. Sagheer, A., Zidan, M., Abdelsamea, M.M.: A novel autonomous perceptron model for pattern classification applications. Entropy 21(8), 763 (2019)

    MathSciNet  Google Scholar 

  33. Singh, R.P.: Generalized Struve’s function and its recurrence relations. Ranchi Univ. Math. J. 5, 67–75 (1974)

    MathSciNet  MATH  Google Scholar 

  34. Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Wiley, New York (1984)

    MATH  Google Scholar 

  35. Tovar, A.A.: Propagation of Laguerre–Bessel–Gaussian beams. JOSA A 17, 2010–2018 (2000)

    MathSciNet  Google Scholar 

  36. Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944)

    MATH  Google Scholar 

  37. Whittaker, E.T.: An expression of certain known functions as generalized hypergeometric functions. Bull. Am. Math. Soc. 10, 125–134 (1903)

    MathSciNet  MATH  Google Scholar 

  38. Xu, P., He, X., Wang, J., Zhan, M.: Trapping a single atom in a blue detuned optical bottle beam trap. Opt. Lett. 35, 2164–2170 (2010)

    Google Scholar 

  39. Zhang, D.W., Yuan, X.-C.: Optical doughnut for optical tweezers. Opt. Lett. 28, 740–742 (2003)

    Google Scholar 

  40. Zidan, M., Abdel-Aty, A.H., El-Sadek, A., Zanaty, E.A., Abdel-Aty, M.: AIP Conf. Proc. 1905, 020005 (2017)

    Google Scholar 

  41. Zidan, M., Abdel-Aty, A., Nguyene, D.M., Mohamed, A.S.A., Al-Sboug, Y., H. leuchh and M. Abdel-Aty,: A quantum algorithm based on entanglement measure for classifying multivariate function into novel hidden classes. Results Phys. 15, 102549 (2019)

  42. Zidan, M., Sagheer, A., Metwally, N.: An autonomous competitive learning algorithm using quantum hamming neural networks. In: Proceedings of the International Joint Conference on Neural Networks (IJCNN, Killarney, Ireland, 2015), IEEE, (2015), pp. 1–7

  43. Zidan, M., Abdel-Aty, A.H., El-shafei, M., Feraig, M., Al-Sbou, Y., Eleuch, H., Abdel-Aty, M.: Quantum classification algorithm based on competitive learning neural network and entanglement measure. Appl. Sci. 9(7), 1277 (2019)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LPNAMME, Laser Physics Group, Department of Physics, Faculty of Sciences, Chouaïb Doukkali University, P. B 20, 24000, El Jadida, Morocco

    A. Belafhal & E. M. El Halba

  2. Department of Mathematics, School of Basic and Applied Sciences, Lingaya’s Vidyapeeth, Faridabad, Haryana, 121002, India

    T. Usman

Authors
  1. A. Belafhal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. M. El Halba
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. Usman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Usman.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belafhal, A., El Halba, E.M. & Usman, T. An Integral Transform Involving the Product Of Bessel Functions and Whittaker Function and Its Application. Int. J. Appl. Comput. Math 6, 177 (2020). https://doi.org/10.1007/s40819-020-00930-2

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-020-00930-2

Keywords

Mathematics Subject Classification

Search

Navigation