Research article

Some integrals involving Coulomb functions associated with the three-dimensional proper Lorentz group

  • Received: 01 May 2020 Accepted: 29 June 2020 Published: 02 July 2020
  • MSC : 33C10, 33C80, 33B15, 33C05

  • For two continual bases in the representation space, we obtain the matrix elements of the linear operator transforming the first basis into the second. These elements are expressed in terms of Coulomb wave functions. Computing the matrix elements of subrepresentations to some subgroups or their separate elements and using the connection between above bases, we evaluate some integrals involving Coulomb wave functions.

    Citation: I. A. Shilin, Junesang Choi, Jae Won Lee. Some integrals involving Coulomb functions associated with the three-dimensional proper Lorentz group[J]. AIMS Mathematics, 2020, 5(6): 5664-5682. doi: 10.3934/math.2020362

    Related Papers:

  • For two continual bases in the representation space, we obtain the matrix elements of the linear operator transforming the first basis into the second. These elements are expressed in terms of Coulomb wave functions. Computing the matrix elements of subrepresentations to some subgroups or their separate elements and using the connection between above bases, we evaluate some integrals involving Coulomb wave functions.


    加载中


    [1] G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
    [2] Z. Bentalha, Representations of the Coulomb matrix elements by means of Appell hypergeometric function F2, Math. Phys. Anal. Geom., 21 (2018), 10.
    [3] J. W. Brown and R. V. Churchill, Complex Variables and Applications, Sixth Edi., McGraw-Hill International Editions, 1996.
    [4] A. R. Curtis, Coulomb Wave Functions, Cambridge University Press, Cambridge, 1964.
    [5] A. Dzieciol, S. Yngve and P. O. Froman, Coulomb wave functions with complex values of the variable and the parameters, J. Math. Phys., 40 (1999), 6145-6166. doi: 10.1063/1.533083
    [6] A. Erdélyi, M. Kennedy and J. L. McGregory, Asymptotic Forms of Coulomb Wave Functions, Part I, California Institute of Technology, Pasadena, 1955.
    [7] A. Erdélyi, W. Magnus, F. Oberhettinger, et al. Higher Transcendental Functions, McGraw-Hill Book Company, New York, Toronto and London, 1953.
    [8] D. Gaspard, Connection formulas between Coulomb wave functions, J. Math. Phys., 59 (2018), 112104.
    [9] I. M. Gel'fand and G. E. Shilov, Generalized Functions: AMS Chelsea Publishing, 2016.
    [10] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 2007.
    [11] D. Hartree, The Calculation of Atomic Structures, Ney York, Wiley, 1957.
    [12] N. Hayek, Estudio de la ecuación diferencial xy'' + (ν+ 1) y' + y = 0 y de sus aplicaciones, Collect. Math., 18 (1966-1967), 57-174.
    [13] N. Hayek, Sobre la transformación de Hankel, in Actas de la VIII Reunión Anual de Matemáticos Epañoles, (1967), 47-60.
    [14] H. Jeffreys and B. Jeffreys, Methods of Mathematical Phisics, Cambridge University Press, Cambridge, 1956.
    [15] D. A. Morales, On the evaluation of integrals with Coulomb Sturmian radial functions, J. Math. Chem., 54 (2016), 682-689. doi: 10.1007/s10910-015-0588-1
    [16] J. M. R. Méndez Pérez and M. M. Socas Robayna, A pair of generalized Hankel-Clifford transformations and their applications, J. Math. Anal. Appl., 154 (1991), 543-557. doi: 10.1016/0022-247X(91)90057-7
    [17] A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics, Birkhauser, Basel, Boston, 1988.
    [18] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, et al. NIST Handbook of Mathematical Functions hardback and CD-ROM, Cambridge University Press, 2010.
    [19] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 1: Elementary Functions, OPA (Overseas Publishers Association), Amsterdam B. V. Published under the license of Gordon and Breach Science Publishers, 1986.
    [20] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 2: Special Functions, OPA (Overseas Publishers Association), Amsterdam B. V. Published under the license of Gordon and Breach Science Publishers, 1986.
    [21] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 3: More Special Functions, OPA (Overseas Publishers Association), Amsterdam B. V. Published under the license of Gordon and Breach Science Publishers, 1986.
    [22] E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
    [23] I. A. Shilin, On some integral transforms of Coulomb functions related to three-dimensional proper Lorentz group, arXiv: 1904.03729.
    [24] I. A. Shilin and J. Choi, Certain relations between Bessel and Whittaker functions related to some diagonal and block-diagonal 3×3-matrices, J. Nonlinear Sci. Appl., 10 (2017), 560-574. doi: 10.22436/jnsa.010.02.20
    [25] I. A. Shilin and J. Choi, Certain connections between the spherical and hyperbolic bases on the cone and formulas for related special functions, Integr. Transf. Spec. F., 25 (2014), 374-383. doi: 10.1080/10652469.2013.860454
    [26] I. A. Shilin and J. Choi, Some connections between the spherical and parabolic bases on the cone expressed in terms of the Macdonald function, Abs. Appl. Anal., 2014 (2014), 741079.
    [27] H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
    [28] H. M. Srivastava, H. A. Mavromatis and R. S. Alassar, Remarks on some associated Laguerre integral results, Appl. Math. Lett., 16 (2003), 1131-1136. doi: 10.1016/S0893-9659(03)90106-6
    [29] I. J. Thompson and A. R. Barnett, Coulomb and Bessel functions of complex arguments and order, J. Comput. Phys., 64 (1986), 490-509. doi: 10.1016/0021-9991(86)90046-X
    [30] N. J. Vilenkin and A. U. Klimyk, Representations of Lie Groups and Special Functions, Vol. 2, Dordrecht, Kluwer Academic Publishers, 1993.
    [31] N. J. Vilenkin and M. A. Sleinikova, Integral relations for the Whittakers functions and the representations of the three-dimensional Lorentz group, Mathematics of the USSR-Sbornik, 10 (1970), 173-180. doi: 10.1070/SM1970v010n02ABEH001593
    [32] R. F. Wehrhahn, Y. F. Smirnov and A. M. Shirokov, Symmetry scattering on the hyperboloid S O(2, 1)/S O(2) in different coordinate systems, J. Math. Phys., 33 (1992), 2384-2389. doi: 10.1063/1.529979
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2847) PDF downloads(245) Cited by(4)

Article outline

Figures and Tables

Figures(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog