Elementary evaluation of certain infinite integrals involving Bessel functions
Authors:
V. I. Fabrikant and G. Dôme
Journal:
Quart. Appl. Math. 59 (2001), 1-24
MSC:
Primary 33C10; Secondary 31B05, 33C05
DOI:
https://doi.org/10.1090/qam/1811092
MathSciNet review:
MR1811092
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Although it is known theoretically that certain infinite integrals of Bessel functions can be expressed in terms of elementary functions, the practical evaluation of such integrals was quite difficult due to the algebraic complexity of the expressions involved. A simple and elegant algebra is introduced here which allows these integrals to be calculated in an elementary way in terms of elementary functions. Some relationships are shown between the integrals involving Bessel functions and two-dimensional integrals over a circle of elementary functions involving distances between points. A comparison is made with existing results, and some of them were found in error (or were misprints).
- Alexander Apelblat, Table of definite and infinite integrals, Physical Sciences Data, vol. 13, Elsevier Scientific Publishing Co., Amsterdam, 1983. MR 902582
A. Erdélyi, ed., W. Magnus, F. Oberhettinger, and F. C. Tricomi, Table of Integral Transforms, vol. I, McGraw-Hill, New York, 1953
- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0061695
- H. A. Elliott, Axial symmetric stress distributions in aeolotropic hexagonal crystals. The problem of the plane and related problems, Proc. Cambridge Philos. Soc. 45 (1949), 621–630. MR 32414, DOI https://doi.org/10.1017/s0305004100025305
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 5th ed., Academic Press, Inc., Boston, MA, 1994. Translation edited and with a preface by Alan Jeffrey. MR 1243179
- 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
- V. I. Fabrikant, Mixed boundary value problems of potential theory and their applications in engineering, Mathematics and its Applications, vol. 68, Kluwer Academic Publishers Group, Dordrecht, 1991. MR 1175754
- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 1, Gordon & Breach Science Publishers, New York, 1986. Elementary functions; Translated from the Russian and with a preface by N. M. Queen. MR 874986
- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 2, Gordon & Breach Science Publishers, New York, 1986. Special functions; Translated from the Russian by N. M. Queen. MR 874987
- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 3, Gordon and Breach Science Publishers, New York, 1990. More special functions; Translated from the Russian by G. G. Gould. MR 1054647
- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 4, Gordon and Breach Science Publishers, New York, 1992. Direct Laplace transforms. MR 1162979
- 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
A. Apelblat, Table of Definite and Infinite Integrals, Elsevier, Amsterdam, 1983
A. Erdélyi, ed., W. Magnus, F. Oberhettinger, and F. C. Tricomi, Table of Integral Transforms, vol. I, McGraw-Hill, New York, 1953
A. Erdélyi, ed., W. Magnus, F. Oberhettinger, and F. C. Tricomi, Table of Integral Transforms, vol. II, McGraw-Hill, New York, 1954
H. A. Elliott, Axial symmetric stress distributions in aeolotropic hexagonal crystals. The problem of the plane and related problems, Proc. Cambridge Phil. Soc. 45, 621–630 (1949)
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th edition, A. Jeffrey, ed., Academic Press, New York, 1994
V. I. Fabrikant, Application of Potential Theory in Mechanics. Selection of New Results, Kluwer, Dordrecht, 1989
V. I. Fabrikant, Mixed Boundary Value Problems of Potential Theory and their Application in Engineering, Kluwer, Dordrecht, 1991
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, vol. 1, Elementary Functions, Gordon and Breach, New York, 1986
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, vol. 2, Special Functions, Gordon and Breach, New York, 1986
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, vol. 3, More Special Functions, Gordon and Breach, New York, 1990
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, vol. 4, Direct Laplace Transforms, Gordon and Breach, New York, 1992
G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., University Press, Cambridge, 1966
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
33C10,
31B05,
33C05
Retrieve articles in all journals
with MSC:
33C10,
31B05,
33C05
Additional Information
Article copyright:
© Copyright 2001
American Mathematical Society
