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Neumann Series

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Series of Bessel and Kummer-Type Functions

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2207))

Abstract

The goal of present chapter is to study in details the integral representations of the Neumann series (of the first and second type) of Bessel and modified Bessel functions of the first and second kind. In order to achieve our goal we use several methods: the Euler–Maclaurin summation technique, differential equation technique, fractional integration technique. Moreover, we present some interesting results on the coefficients of Neumann series, product of modified Bessel functions of the first and second kind and the cumulative distribution function of the non-central χ 2-distribution.

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Notes

  1. 1.

    Watson remarked that all four formulae that were cited by him [333, 16.53 Eqs. (1), (2), (11), (12)] had been derived by von Lommel (cf. von Lommel’s memoirs [324, 325] for further details).

  2. 2.

    It is worth to mention here that the above procedure for modified Bessel functions is similar to the method for Bessel functions applied by Wilkins [335]. See also Andrews et al. [7] for more details. More precisely, Wilkins proved that the Hankel functions \(\big (H_{\nu }^{(1)}\big )^2\) and \(\big (H_{\nu }^{(2)}\big )^2,\) as well as \(J_{\nu }^2+Y_{\nu }^2,\) where J ν and Y ν stand for the Bessel functions of the first and second kind, are particular solutions of the third order homogeneous differential equation [7, p. 225]

    $$\displaystyle \begin{aligned} x^2y'''(x)+3xy''(x)+(1+4x^2-4\nu^2)y'(x)+4xy(x)=0. \end{aligned}$$

    The above result was used to prove the celebrated Nicholson formula [7, p. 224]

    $$\displaystyle \begin{aligned}J_{\nu}^2(x)+Y_{\nu}^2(x)=\frac{8}{\pi^2}\int_0^{\infty}K_0(2x\sinh t)\cosh(2\nu t) \mathrm{d}t,\end{aligned}$$

    which generalizes the trigonometric identity \(\sin ^2x+\cos ^2x=1.\)

References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1965)

    Google Scholar 

  2. Agrest, M.M., Maksimov, M.S.: Theory of Incomplete Cylindrical Functions and their Applications. Springer, New York (1971)

    Google Scholar 

  3. Al-Jarrah, A., Dempsey, K.M., Glasser, M.L.: Generalized series of Bessel functions. J. Comput. Appl. Math. 143, 1–8 (2002)

    Google Scholar 

  4. Al-Salam, W.A.: A generalized Turán expression for Bessel functions. Am. Math. Mon. 68(2), 146–149 (1961)

    Google Scholar 

  5. András, S., Baricz, Á., Sun, Y.: The generalized Marcum Q-function: an orthogonal polynomial approach. Acta Univ. Sapientiae Math. 3(1), 60–76 (2011)

    Google Scholar 

  6. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and it Applications, vol. 71. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  7. Bailey, W.N.: Generalized Hypergeometric Series. Cambridge Tract, vol. 32. Cambridge University Press, Cambridge (1935)

    Google Scholar 

  8. Baricz, Á.: On a product of modified Bessel functions. Proc. Am. Math. Soc. 137(1), 189–193 (2009)

    Google Scholar 

  9. Baricz, Á.: Bounds for modified Bessel functions of the first and second kind. Proc. Edin. Math. Soc. 53(3), 575–599 (2010)

    Google Scholar 

  10. Baricz, Á.: Generalized Bessel functions of the first kind. Lecture Notes in Mathematics, vol. 1994. Springer, Berlin (2010)

    Google Scholar 

  11. Baricz, Á., Pogány, T.K.: Properties of the product of modified Bessel functions. In: Milovanović, G.V., Rassias, M.Th. (eds.) Analytic Number Theory, Approximation Theory, and Special Functions, pp. 809–820. Springer, Berlin (2014). In Honor of Hari M. Srivastava

    Google Scholar 

  12. Baricz, Á., Pogány, T.K.: Turán determinants of Bessel functions. Forum Math. 26(1), 295–322 (2014)

    Google Scholar 

  13. Baricz, Á., Ponnusamy, S.: On Turán type inequalities for modified Bessel functions. Proc. Am. Math. Soc. 141(2), 523–532 (2013)

    Google Scholar 

  14. Baricz, Á., Jankov, D., Pogány, T.K.: Integral representations for Neumann-type series of Bessel functions I ν , Y ν and K ν . Proc. Am. Math. Soc. 140(3), 951–960 (2012)

    Google Scholar 

  15. Baricz, Á., Jankov, D., Pogány, T.K.: Neumann series of Bessel functions. Integral Transforms Spec. Funct. 23(7), 529–538 (2012)

    Google Scholar 

  16. Baricz, Á., Jankov, D., Pogány, T.K.: Turán type inequalities for Krätzel functions. J. Math. Anal. Appl. 388(2), 716–724 (2012)

    Google Scholar 

  17. Brychkov, Yu.A.: On some properties of the Marcum Q function. Integral Transforms Spec. Funct. 23(3), 177–182 (2012)

    Google Scholar 

  18. Chaudhry, M.A., Zubair, S.M.: Generalized incomplete gamma function with applications. J. Comput. Appl. Math. 55, 99–124 (1994)

    Google Scholar 

  19. Chessin, A.S.: Sur l’équation de Bessel avec second membre. Compt. Rend. 135, 678–679 (1902)

    Google Scholar 

  20. Chessin, A.S.: Sur une classe d’équations différentielles réductibles a l’équation de Bessel. Compt. Rend. 136, 1124–1126 (1903)

    Google Scholar 

  21. Cochran, J.A.: The monotonicity of modified Bessel functions with respect to their order. J. Math. Phys. 46, 220–222 (1967)

    Google Scholar 

  22. De Micheli, E.: Integral representation for Bessel’s functions of the first kind and Neumann series (2017). arXiv:1708.09715v1 [math.CA]

    Google Scholar 

  23. Delfino, F., Procopio, R., Rossi, M.: Evaluation of capacitance matrix of a finite-length multiconductor transmission line. IEE Proc.: Sci. Meas. Technol. 151, 347–353 (2004)

    Google Scholar 

  24. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 2. McGraw-Hill, New York, Toronto, London (1953)

    Google Scholar 

  25. Fejzullahu, B.Xh.: Neumann series and Lommel functions of two variables. Integral Transforms Spec. Funct. 27(6), 443–453 (2016)

    Google Scholar 

  26. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 6th edn. Academic, San Diego, CA (2000)

    Google Scholar 

  27. Graham, R.L.: Application of the FKG inequality and its relatives. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming: The State of the Art, pp. 115–131. Springer, Berlin (1983)

    Google Scholar 

  28. Grandison, S., Penfold, R., Vanden-Broeck, J.M.: A rapid boundary integral equation technique for protein electrostatics. J. Comput. Phys. 224, 663–680 (2007)

    Google Scholar 

  29. Gröbner, W., Hofreiter, N.: Integraltafel: Zweiter Teil. Bestimmte Integrale. Springer, Wien (1973)

    Google Scholar 

  30. Hansen, E.R.: A Table of Series and Products. Prentice-Hall, Englewood Cliffs, NJ (1975)

    Google Scholar 

  31. Hantush, M.S., Jacob, C.E.: Non-steady radial flow in an infinite leaky aquifer. Trans. Am. Geophys. Union 36, 95–100 (1955)

    Google Scholar 

  32. Hasan, A.A.: Electrogravitational stability of oscillating streaming fluid cylinder. Phys. B. 406, 234–240 (2011)

    Google Scholar 

  33. http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F2/03/03/07/09/0004/ (2016). Accessed 31 Oct 2016

  34. http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F2/03/03/09/11/0003/ (2016). Accessed 31 Oct 2016

  35. http://functions.wolfram.com/07.20.07.0002.01 (2016). Accessed 31 Oct 2016

  36. http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/20/01/02/0004/ (2016). Accessed 31 Oct 2016

  37. http://mathworld.wolfram.com/MarcumQ-Function.html (2016). Accessed 31 Oct 2016

  38. Ismail, M.E.H.: Complete monotonicity of modified Bessel functions. Proc. Am. Math. Soc. 108(2), 353–361 (1990)

    Google Scholar 

  39. Jankov, D., Pogány, T.K., Süli, E.: On the coefficients of Neumann series of Bessel functions. J. Math. Anal. Appl. 380(2), 628–631 (2011)

    Google Scholar 

  40. Jankov Maširević, D.: On new formulas for the cumulative distribution function of the noncentral chi-square distribution. Mediterr. J. Math. 14(2), Art 66, 13 pp. (2017)

    Google Scholar 

  41. Jankov Maširević, D., Pogány, T.K.: New summations of Neumann series of modified Bessel functions. J. Anal. 23, 47–57 (2015)

    Google Scholar 

  42. Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, vol. 2. Wiley, New York (1995)

    Google Scholar 

  43. Jones, A.L.: An extension of an inequality involving modified Bessel functions. J. Math. Phys. 47, 220–221 (1968)

    Google Scholar 

  44. Karamata, J.: Theory and Applications of Stieltjes integral. Srpska Akademija Nauka, Posebna izdanja CLIV, Matematički institut, Knjiga I, Beograd (1949) (in Serbian)

    Google Scholar 

  45. Karatsuba, E.A., Moretti, P.: Inversion time of large spins. J. Math. Phys. 46(4), 042101:1–7 (2005)

    Google Scholar 

  46. Klimek, S., McBride, M.: Global boundary conditions for a Dirac operator on the solid torus. J. Math. Phys. 52, Article 063518, 14 pp. (2011)

    Google Scholar 

  47. Kravchenko, V.V., Torba, S.M.: A Neumann series of Bessel functions representation for solutions of Sturm-Liouville equations (2016). arXiv:1612.08803v1 [math.CA]

    Google Scholar 

  48. Kravchenko, V.V., Torba, S.M., Castillo-Prez, R.: A Neumann series of Bessel functions representation for solutions of perturbed Bessel equations. Appl. Anal. (2017). 10.1080/00036811.2017.1284313

    Google Scholar 

  49. Kravchenko, V.V., Navarro, L.J., Torba, S.M.: Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions. Appl. Math. Comput. 314, 173–192 (2017)

    Google Scholar 

  50. Laforgia, A.: Bounds for modified Bessel functions. J. Comput. Appl. Math. 34(3), 263–267 (1991)

    Google Scholar 

  51. Lin, S.D., Shyu, J.C., Nishimoto, K., Srivastava, H.M.: Explicit solutions of some general families of ordinary and partial differential equations associated with the Bessel equation by means of fractional calculus. J. Fract. Calc. 25, 33–45 (2004)

    Google Scholar 

  52. Lin, S.D., Ling, W.C., Nishimoto, K., Srivastava, H.M.: A simple fractional-calculus approach to the solutions of the Bessel differential equation of general order and some of its applications. Comput. Math. Appl. 49(9–10), 1487–1498 (2005)

    Google Scholar 

  53. Luke, Y.L.: Integrals of Bessel Functions. McGraw-Hill, New York-Toronto-London (1962)

    Google Scholar 

  54. Marcum, J.I.: A statistical theory of target detection by pulsed radar. IRE Trans. Inf. Theory 6, 59–267 (1960)

    Google Scholar 

  55. Martin, P.A.: Acoustic waves in slender axisymmetric tubes. J. Sound Vib. 286, 55–68 (2005)

    Google Scholar 

  56. Martin, P.A., Berger, J.R.: Waves in wood: free vibrations of a wooden pole. J. Mech. Phys. Solids 49, 1155–1178 (2001)

    Google Scholar 

  57. Maximon, L.C.: On the representation of indefinite integrals containing Bessel functions by simple Neumann series. Proc. Am. Math. Soc. 7(6), 1054–1062 (1956)

    Google Scholar 

  58. Mei, Z., Zhao, D., Gu, J.: Comparison of two approximate methods for hard-edged diffracted flat-topped light beams. Opt. Commun. 267, 58–64 (2006)

    Google Scholar 

  59. Meligy, A.S.: On Whittaker and Coulomb functions. J. Lond. Math. Soc. 37, 141–144 (1962)

    Google Scholar 

  60. Morales-Jimenez, D., Lopez-Martinez, F.J., Martos-Naya, E., Paris, J.F., Lozano, A.: Connections between the generalized Marcum Q-function and a class of hypergeometric functions. IEEE Trans. Inform. Theory 60(2), 1077–1082 (2014)

    Google Scholar 

  61. Nadon, M., Campbell, L.J.: An exact expression for transient forced internal gravity waves in a Boussinesq fluid. Wave Motion 44, 340–345 (2007)

    Google Scholar 

  62. Neumann, C.G.: Theorie der Besselschen Funktionen. B.G. Teubner, Leipzig (1867)

    Google Scholar 

  63. Newberger, B.S.: New sum rule for products of Bessel functions with application to plasma physics. J. Math. Phys. 23(7), 1278–1281 (1982)

    Google Scholar 

  64. Novomestky, F.: Asymptotic expression for the unit-step and dirac delta functions. SIAM J. Appl. Math. 27(4), 521–525 (1974)

    Google Scholar 

  65. Oberhettinger, F.: Tables of Bessel Transforms. Springer, New York (1972)

    Google Scholar 

  66. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. NIST and Cambrigde University Press, Cambridge (2010)

    Google Scholar 

  67. Patnaik, P.B.: The non-central χ 2- and the F-distributions and their applications. Biometrika 36, 202–232 (1949)

    Google Scholar 

  68. Pearson, E.S.: Note on an approximation to the distribution of noncentral χ 2. Biometrika 46, 364–364 (1959)

    Google Scholar 

  69. Penfold, R., Vanden-Broeck, J.M., Grandison, S.: Monotonicity of some modified Bessel function products. Integral Transforms Spec. Funct. 18, 139–144 (2007)

    Google Scholar 

  70. Perron, O.: Zur Theorie der Dirichletschen Reihen. J. Reine Angew. Math. 134, 95–143 (1908)

    Google Scholar 

  71. Phillips, R.S., Malin, H.: Bessel function approximations. Am. J. Math. 72, 407–418 (1950)

    Google Scholar 

  72. Pogány, T.K., Süli, E.: Integral representation for Neumann series of Bessel functions. Proc. Am. Math. Soc. 137(7), 2363–2368 (2009)

    Google Scholar 

  73. Pogány, T.K., Srivastava, H.M., Tomovski, ž.: Some families of Mathieu a–series and alternating Mathieu a-series. Appl. Math. Comput. 173(1), 69–108 (2006)

    Google Scholar 

  74. Pogány, T.K., Baricz, Á., Rudas, I.: Incomplete Krätzel function model of leaky aquifer and alike functions. In: Proceedings of the 10th Jubilee IEEE International Symposium on Applied Computational Intelligence and Informatics (May 21–23), Timişoara, Romania, pp. 59–62 (2015)

    Google Scholar 

  75. Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series, vol. 2. Special Functions. Gordon and Breach Science Publishers, New York (1986)

    Google Scholar 

  76. Radwan, A.E., Hasan, A.A.: Magneto hydrodynamic stability of self-gravitational fluid cylinder. Appl. Math. Model. 33, 2121–2131 (2009)

    Google Scholar 

  77. Radwan, A.E., Dimian, M.F., Hadhoda, M.K.: Magnetogravitational stability of a bounded gas-core fluid jet. Appl. Energy 83, 1265–1273 (2006)

    Google Scholar 

  78. Reudink, D.O.: On the signs of the ν-derivatives of the modified Bessel functions I ν (x) and K ν (x). J. Res. Natl. Bur. Stand. B72, 279–280 (1968)

    Google Scholar 

  79. Rice, S.O.: Mathematical analysis of random noise III. Bell Syst. Tech. J. 24, 46–156 (1945)

    Google Scholar 

  80. Robert, C.: Modified Bessel functions and their applications in probability and statistics. Stat. Probab. Lett. 9, 155–161 (1990)

    Google Scholar 

  81. Robinson, N.I.: An isotropic elastic medium containing a cylindrical borehole with a rigid plug. Int. J. Solids Struct. 39, 4889–4904 (2002)

    Google Scholar 

  82. Salem, M.A., Kamel, A.H., Osipov, A.V.: Electromagnetic fields in the presence of an infinite dielectric wedge. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462(2), 2503–2522 (2006)

    Google Scholar 

  83. Sankaran, M.: Approximations to the noncentral chi-square distribution. Biometrika 50, 199–204 (1963)

    Google Scholar 

  84. Siemon, P.: Über die Integrale einer nicht homogenen Differentialgleichung zweiter Ordnung. In: Programm der Luisienschuhle. Sechster Abschnitt. Differential- und Integralrechnung. Capitel 5. Gewöhnliche Differentialgleichungen, Berlin (1890)

    Google Scholar 

  85. Srivastava, H.M., Karlsson, P.W.: Multiple Gaussian Hypergeometric Series. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd./Halsted Press [Wiley], Chichester/New York (1985)

    Google Scholar 

  86. Temme, N.M.: Asymptotic and numerical aspects of the noncentral chi-square distribution. Comput. Math. Appl. 25(5), 55–63 (1993)

    Google Scholar 

  87. Thiruvenkatachar, V.R., Nanjundiah, T.S.: Inequalities concerning Bessel functions and orthogonal polynomials. Proc. Indian Acad. Sci. Sect. A 33, 373–384 (1951)

    Google Scholar 

  88. Van Heijster, P., Sandstede, B.: Planar radial spots in a three-component FitzHugh-Nagumo system. J. Nonlinear Sci. 21, 705–745 (2011)

    Google Scholar 

  89. Van Heijster, P., Doelman, A., Kaper, T.J.: Pulse dynamics in a three-component system: stability and bifurcations. Phys. D. Nonlinear Phenom. 237(24), 3335–3368 (2008)

    Google Scholar 

  90. Van Heijster, P., Doelman, A., Kaper, T.J., Promislow, K.: Front interactions in a three-component system. SIAM J. Appl. Dyn. Syst. 9, 292–332 (2010)

    Google Scholar 

  91. Veling, E.J.M.: The generalized incomplete Gamma function as sum over modified Bessel functions of the first kind. J. Comput. Appl. Math. 235, 4107–4116 (2011)

    Google Scholar 

  92. von Lommel, E.C.J.: 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 

  93. von Lommel, E.C.J.: Die Beugungserscheinungen geradlinig begrenzter Schirme. Abh. der math. phys. Classe der k. b. Akad. der Wiss. (München) 15, 529–664 (1884–1886)

    Google Scholar 

  94. Wang, P.Y.: Solutions of Some Certain Classes of Differential Equations by Means of Fractional Calculus. Ph.D. Dissertation, Department of Applied Mathematics, Chung Yuan Christian University Chung-Li, Taiwan (2006)

    Google Scholar 

  95. Wang, P.Y., Lin, S.D., Srivastava, H.M.: Remarks on a simple fractional-calculus approach to the solutions of the Bessel differential equation of general order and some of its applications. Comput. Math. Appl. 51(1), 105–114 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  96. Wang, P.Y., Lin, S.D., Tu, S.T.: A survey of fractional-calculus approaches to the solutions of the Bessel differential equation of general order. Appl. Math. Comput. 187(1), 544–555 (2007)

    MathSciNet  MATH  Google Scholar 

  97. Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)

    MATH  Google Scholar 

  98. Wilkins Jr., J.E.: Neumann series of Bessel functions. Trans. Am. Math. Soc. 64, 359–385 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  99. Wilkins Jr., J.E.: Nicholson’s integral for \(J_n^2(z)+Y_n^2(z)\). Bull. Am. Math. Soc. 54, 232–234 (1948)

    Google Scholar 

  100. Wilkins Jr., J.E.: Neumann series of Bessel functions II. Trans. Am. Math. Soc. 69, 55–65 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  101. Wilson, E.B., Hilfetry, M.M.: The distribution of chi-square. Proc. Natl. Acad. Sci. 17, 684–688 (1931)

    Article  Google Scholar 

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Authors and Affiliations

  1. John von Neumann Faculty of Informatics, Institute of Applied Mathematics, Óbuda University, Budapest, Hungary

    Árpád Baricz & Tibor K. Pogány

  2. Department of Economics, Babeş–Bolyai University, Cluj–Napoca, Romania

    Árpád Baricz

  3. Department of Mathematics, Josip Juraj Strossmayer University of Osijek, Osijek, Croatia

    Dragana Jankov Maširević

  4. Faculty of Maritime Studies, University of Rijeka, Rijeka, Croatia

    Tibor K. Pogány

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Baricz, Á., Jankov Maširević, D., Pogány, T.K. (2017). Neumann Series. In: Series of Bessel and Kummer-Type Functions. Lecture Notes in Mathematics, vol 2207. Springer, Cham. https://doi.org/10.1007/978-3-319-74350-9_2

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