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

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).

**[1]**Alexander Apelblat,*Table of definite and infinite integrals*, Physical Sciences Data, vol. 13, Elsevier Scientific Publishing Co., Amsterdam, 1983. MR**902582****[2]**A. Erdélyi, ed., W. Magnus, F. Oberhettinger, and F. C. Tricomi,*Table of Integral Transforms*, vol. I, McGraw-Hill, New York, 1953**[3]**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****[4]**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**0032414****[5]**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****[6]**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****[7]**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****[8]**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****[9]**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****[10]**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****[11]**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****[12]**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**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
33C10,
31B05,
33C05

Retrieve articles in all journals with MSC: 33C10, 31B05, 33C05

Additional Information

DOI:
https://doi.org/10.1090/qam/1811092

Article copyright:
© Copyright 2001
American Mathematical Society