On the Crossed Term Integral Occuring in the Coulomb Self-Energy of Uniformly Charged Hollow Cylinder

  • Árpád Baricz
  • Tibor K. PogányEmail author
Part of the Topics in Intelligent Engineering and Informatics book series (TIEI, volume 14)


The article aims at studying elliptic integral form of the so–called crossed term integral occurring in description of Coulomb self-energy of a uniformly charged three-dimensional hollow cylinder. Firstly, uniform bounds are established for the Kampé de Fériet double hypergeometric function occurring in the recent related results by Batle, Ciftja and Pogány [4]. Secondly, related bilateral bounding inequalities are obtained for the expressions established in terms of elliptic integrals.


Generalized hypergeometric function Kampé de Fériet hypergeometric function of two variables Bessel function of the first kind Elliptic integrals of the first and second kind Generalized hypergeometric function \({}_pF_q\) 

2010 Mathematics Subject Classification

33C10 33C20 33C65 33C75 33E05 


  1. 1.
    P. Appell, J. Kampé de Fériet, Fonctions hypergeometrique. Polynomes d’Hermite (Gautier–Villars, Paris, 1926)zbMATHGoogle Scholar
  2. 2.
    G.B. Arfken, H.J. Weber, Mathematical Methods for Physicists, 5th edn. (Academic Press, San Diego, 2001)zbMATHGoogle Scholar
  3. 3.
    Á. Baricz, P.L. Butzer, T.K. Pogány, Alternating mathieu series, Hilbert-Eisenstein series and their generalized omega functions, in Analytic Number Theory, Approximation Theory, and Special Functions - In Honor of Hari M. Srivastava, ed. by T. Rassias, G.V. Milovanović (Springer, New York, 2014), pp. 775–808CrossRefGoogle Scholar
  4. 4.
    J. Batle, O. Ciftja, T.K. Pogány, Hypergeometric solutions for Coulomb self-energy model of uniformly charged hollow cylinder. Integral Transforms Spec. Funct. (2019).
  5. 5.
    J. Batle, O. Ciftja, S. Abdalla, M. Elhoseny, M. Alkhambashi, A. Farouk, Equilibrium charge distribution on a finite straight one-dimensional wire. Eur. J. Phys. 38, 055202 (2017)CrossRefGoogle Scholar
  6. 6.
    O. Ciftja, Coulomb self-energy of a uniformly charged three-dimensional cylinder. Phys. B 407, 2803–2807 (2012)CrossRefGoogle Scholar
  7. 7.
    O. Ciftja, Calculation of the Coulomb electrostatic potential created by of a uniformly charged square on its plane: exact mathematical formulas. J. Electrost. 71, 102–108 (2013)CrossRefGoogle Scholar
  8. 8.
    O. Ciftja, A. Babineaux, N. Hafeez, The electrostatic potential of a uniformly charged ring. Eur. J. Phys. 30, 623–627 (2009)CrossRefGoogle Scholar
  9. 9.
    D.C. Giancoli, Physics for Scientists and Engineers, 3rd edn. (Prentice Hall, Englewood Cliffs, 2000)Google Scholar
  10. 10.
    R.H. Good, Comment on ’Charge density on a conducting needle’. Am. J. Phys. 65, 155–156 (1997)CrossRefGoogle Scholar
  11. 11.
    R.H. Good, Classical Electromagnetism (Saunders College Publishing, Philadelphia, 1999)Google Scholar
  12. 12.
    I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 6th edn. (Academic Press, San Diego, 2000)zbMATHGoogle Scholar
  13. 13.
    D.J. Griffiths, Introduction to Electrodynamics, 3rd edn. (Prentice Hall, Englewood Cliffs, 1999)Google Scholar
  14. 14.
    D.J. Griffiths, Y. Li, Charge density on a conducting needle. Am. J. Phys. 64, 706–714 (1996)CrossRefGoogle Scholar
  15. 15.
    J.D. Jackson, Classical Electrodynamics. Fifth Printing (Wiley, New York, 1966)Google Scholar
  16. 16.
    J.D. Jackson, Charge density on thin straight wire, revisited. Am. J. Phys. 68, 789–799 (2000)CrossRefGoogle Scholar
  17. 17.
    J.D. Jackson, Charge density on a thin straight wire: the first visit. Am. J. Phys. 70, 409–410 (2002)CrossRefGoogle Scholar
  18. 18.
    E. Kausel, M.M.I. Baig, Laplace transform of products of Bessel functions: a visitation of earlier formulas. Quart. Appl. Math. 70, 77–97 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    I. Krasikov, Approximations for the Bessel and Airy functions with an explicit error term. LMS J. Comput. Math. 17(1), 209–225 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    I. Krasikov, On the Bessel function \(J_{\nu }(x)\) in the transition region. LMS J. Comput. Math. 17(1), 273–281 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    L. Landau, Monotonicity and bounds on Bessel functions, in Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory (Berkeley, CA, 1999), pp. 147–154. Electronic journal of differential equations, vol. 4 (Southwest Texas State University, San Marcos, 2000)Google Scholar
  22. 22.
    Y.L. Luke, Integrals of Bessel Functions (McGraw-Hill Book Company, New York, 1962)zbMATHGoogle Scholar
  23. 23.
    S. Minakshisundaram, O. Szász, On absolute convergence of multiple Fourier series. Trans. Am. Math. Soc. 61(1), 36–53 (1947)MathSciNetCrossRefGoogle Scholar
  24. 24.
    A.Ya. Olenko, Upper bound on \(\sqrt{x}J_\nu (x)\) and its applications. Integr. Transforms Spec. Funct. 17(6), 455–467 (2006)CrossRefGoogle Scholar
  25. 25.
    R.K. Parmar, T.K. Pogány, Extended Srivastava’s triple hypergeometric \(H_{A,p,q}\) function and related bounding inequalities. J. Contemp. Math. Anal. 52(6), 261–272 (2017); Izvestiya Natsional’noj Akademii Nauk Armenii, Matematika  52(6), 48–63 (2017)Google Scholar
  26. 26.
    G.L. Pollack, D.R. Stump, Electromagnetism (Addison Wesley, Menlo Park, 2002)Google Scholar
  27. 27.
    W.M. Saslow, Electricity, Magnetism and Light (Academic Press, New York, 2002)Google Scholar
  28. 28.
    R.A. Serway, J.W. Jewett Jr., Physics for Scientists and Engineers with Modern Physics, 6th edn. (Brooks/Cole-Thomson Learning, Belmont, 2004)Google Scholar
  29. 29.
    H.M. Srivastava, M.C. Daoust, A note on the convergence of Kampé de Fériet double hypergeometric series. Math. Nachr. 53, 151–159 (1972)MathSciNetCrossRefGoogle Scholar
  30. 30.
    H.M. Srivastava, R. Panda, An integral representation for the product of two Jacobi polynomials. J. Lond. Math. Soc. 2(12), 419–425 (1976)MathSciNetCrossRefGoogle Scholar
  31. 31.
    H.M. Srivastava, T.K. Pogány, Inequalities for a unified Voigt functions in several variables. Russ. J. Math. Phys. 14(2), 194–200 (2007)MathSciNetCrossRefGoogle Scholar
  32. 32.
    E.C.J. von Lommel, Die Beugungserscheinungen einer kreisrunden Öffnung und eines kreisrunden Schirmchens theoretisch und experimentell bearbeitet. Abh. der math. phys. Classe der k. b. Akad. der Wiss. (München) 15, 229–328 (1884–1886)Google Scholar
  33. 33.
    E.C.J. von Lommel, Die Beugungserscheinungen geradlinig begrenzter Schirme. Abh. der math. phys. Classe der k. b. Akad. der Wiss. (München) 15, 529–664 (1884-1886)Google Scholar
  34. 34.
    G.N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1922)zbMATHGoogle Scholar
  35. 35.
    F.R. Zypman, Off-axis electric field of a ring of charge. Am. J. Phys. 74, 295–300 (2006)MathSciNetCrossRefGoogle Scholar
  36. 36.
    H.D. Young, R.A. Freeman, Sears and Zemansky’s University Physics with Modern Physics, 11th edn. (Addison Wesley, Menlo Park, 2004)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institute of Applied MathematicsÓbuda UniversityBudapestHungary
  2. 2.Department of EconomicsBabeş-Bolyai UniversityCluj-NapocaRomania
  3. 3.Faculty od Maritime StudiesUniversity of RijekaRijekaCroatia

Personalised recommendations