Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 

 

The evaluation of certain infinite integrals involving products of Bessel functions: a correlation of formula


Authors: Mark T. Hanson and Igusti W. Puja
Journal: Quart. Appl. Math. 55 (1997), 505-524
MSC: Primary 33C10; Secondary 31B10, 33C75, 73C35
DOI: https://doi.org/10.1090/qam/1466145
MathSciNet review: MR1466145
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This analysis evaluates certain infinite integrals continuing products of Bessel functions of integer order, an exponential and a power. The integrals considered here have been previously evaluated in the literature in two different forms. In one instance they have been written in terms of complete elliptic integrals of the first, second, and third kind. Some of these integrals have also been evaluated in terms of a Legendre function of the second kind and a complete elliptic integral of the third kind. A recent result in elasticity obtained by the authors has led to a new form for the evaluations of these integrals. The integrals are still evaluated in terms of complete elliptic integrals; however, a new modulus (and parameter for the complete elliptic integral of the third kind) is used. The new form used for the complete elliptic integral of the third kind allows the integral evaluations to be written in a more convenient form than previously given. The new form for the complete elliptic integral of the third kind is also utilized in the evaluations using the Legendre function of the second kind. The new forms to the integral evaluations derived presently are correlated with existing results in the literature.


References [Enhancements On Off] (What's this?)

  • [1] Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. MR 0277773
  • [2] G. Eason, B. Noble, and I. N. Sneddon, On certain integrals of Lipschitz-Hankel type involving products of Bessel functions, Philos. Trans. Roy. Soc. London. Ser. A. 247 (1955), 529–551. MR 0069961, https://doi.org/10.1098/rsta.1955.0005
  • [3] A. Erdelyi, Tables of Integral Transforms, Vol. 2, McGraw-Hill, 1954, p. 50
  • [4] V. I. Fabrikant, Applications of potential theory in mechanics, Mathematics and its Applications, vol. 51, Kluwer Academic Publishers Group, Dordrecht, 1989. A selection of new results. MR 1042755
  • [5] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed., Academic Press, Inc., San Diego, CA, 2000. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. MR 1773820
  • [6] Mark T. Hanson and Igusti W. Puja, Love’s circular patch problem revisited: closed form solutions for transverse isotropy and shear loading, Quart. Appl. Math. 54 (1996), no. 2, 359–384. MR 1388022, https://doi.org/10.1090/qam/1388022
  • [7] Hisao Hasegawa, Ven-Gen Lee, and Toshio Mura, Green’s functions for axisymmetric problems of dissimilar elastic solids, Trans. ASME J. Appl. Mech. 59 (1992), no. 2, 312–320. MR 1176291, https://doi.org/10.1115/1.2899522
  • [8] H. Hasegawa, V. Lee, and T. Mura, The stress fields caused by a circular cylindrical inclusion, ASME J. Appl. Mech. 59, 312-320 (1992)
  • [9] Carl Heuman, Tables of complete elliptic integrals, J. Math. Phys. Mass. Inst. Tech. 20 (1941), 127–206. MR 0003572, https://doi.org/10.1002/sapm1941201127
  • [10] A. E. H. Love, The stress produced in a semi-infinite solid by pressure on part of the boundary, Philos. Trans. Roy. Soc. London A228, 377-420 (1929)
  • [11] R. Muki, Asymmetric problems of the theory of elasticity for a semi-infinite solid and thick plate, Progress in solid mechanics, Vol. 1, North-Holland Publishing Co., Amsterdam, 1960, pp. 399–439. MR 0113357
  • [12] Ian N. Sneddon, Fourier Transforms, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1951. MR 0041963
  • [13] I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, 1972, pp. 325-352
  • [14] M. R. Šura-Bura, Evaluation of an integral containing a product of Bessel functions, Doklady Akad. Nauk SSSR (N.S.) 73 (1950), 901–903 (Russian). MR 0037403
  • [15] K. Terazawa, On the elastic equilibrium of a semi-infinite solid, J. College Sci., University of Tokyo, Vol. 37, article 7, 1916
  • [16] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 33C10, 31B10, 33C75, 73C35

Retrieve articles in all journals with MSC: 33C10, 31B10, 33C75, 73C35


Additional Information

DOI: https://doi.org/10.1090/qam/1466145
Article copyright: © Copyright 1997 American Mathematical Society


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website