A Matlab Implementation of an Algorithm for Computing Integrals of Products of Bessel Functions

We present a Matlab program that computes infinite range integrals of an arbitrary product of Bessel functions of the first kind. The algorithm uses an integral representation of the upper incomplete Gamma function to integrate the tail of the integrand. This paper describes the algorithm and then focuses on some implementation aspects of the Matlab program. Finally we mention a generalisation that incorporates the Laplace transform of a product of Bessel functions.

[1]  D. E. Amos.  Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order , 1986, TOMS.

[2]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[3]  An Efficient Solution of a Class of Integrals Arising in Antenna Theory , 2002 .

[4]  Ronald Cools,et al.  A stable recurrence for the incomplete gamma function with imaginary second argument , 2006, Numerische Mathematik.

[5]  V. Adamchik.  The evaluation of integrals of Bessel functions via G -function identities , 1995 .

[6]  N. Sonine,et al.  Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries , 1880 .

[7]  M. Tezer,et al.  On the numerical evaluation of an oscillating infinite series , 1989 .

[8]  David M. Miller,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[9]  A. Panagopoulos,et al.  On an integral related to biaxially anisotropic media , 2002 .

[10]  John T. Conway,et al.  Analytical solutions for the Newtonian gravitational field induced by matter within axisymmetric boundaries , 2000 .

[11]  Marina L. Gavrilova,et al.  Computational Science and Its Applications — ICCSA 2003 , 2003 .

[12]  A. A. Pivovarov,et al.  On the evaluation of sunset-type Feynman diagrams , 1998 .

[13]  S.V. Savov,et al.  An efficient solution of a class of integrals arising in antenna theory , 2002, IEEE Antennas and Propagation Magazine.

[14]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[15]  Stan Wagon,et al.  The SIAM 100-Digit Challenge - A study in High-Accuracy Numerical Computing , 2004, The SIAM 100-Digit Challenge.

[16]  H. M. McConnell,et al.  Hydrodynamics of quantized shape transitions of lipid domains , 1995, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[17]  Jose M. Roesset.  Nondestructive Dynamic Testing of Soils and Pavements , 1998 .

[18]  Anthony M. J. Davis.  Drag modifications for a sphere in a rotational motion at small, non-zero Reynolds and Taylor numbers : wake interference and possibly Coriolis effects , 1992 .

[19]  Toru Mogi,et al.  Electromagnetic Response of a Large Circular Loop Source on a Layered Earth: A New Computation Method , 2005 .

[20]  S. Davis.  Scalar field theory and the definition of momentum in curved space , 2001 .

[21]  Raman Kashyap,et al.  Physical properties of optical fiber sidetap grating filters: free-space model , 1999 .

[22]  S. K. Lucas,et al.  Evaluating infinite integrals involving products of Bessel functions of arbitrary order , 1995 .

[23]  Howard A. Stone,et al.  Motion of a rigid particle in a rotating viscous flow: an integral equation approach , 1994, Journal of Fluid Mechanics.

[24]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[25]  Ruelle classical resonances and dynamical chaos: The three- and four-disk scatterers. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[26]  Serge Winitzki.  Computing the Incomplete Gamma Function to Arbitrary Precision , 2003, ICCSA.

[27]  Peter Vary,et al.  Multichannel Direction-Independent Speech Enhancement Using Spectral Amplitude Estimation , 2003, EURASIP J. Adv. Signal Process..

[28]  J. Borwein.  The SIAM 100-Digit challenge: a study in high-accuracy numerical computing , 1987 .