Evaluating infinite integrals involving products of Bessel functions of arbitrary order

Lucas, Stephen K.

doi:10.1016/0377-0427(95)00143-3

Cited by 79 publications

(50 citation statements)

References 8 publications

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“…Here, the resulting functions g 1 , g 2 are much more complicated and involve combinations of Bessel's functions of first and second kind [33]. However, the relevant fact is its simple asymptotic behavior, so WA can be applied to them.…”

Section: Extension To Products Of Oscillating Functionsmentioning

confidence: 99%

“…To witness, WA has been applied to the benchmark integral introduced and accurately computed by Lucas [33]: Figure 3 compares the quality of the results obtained with the Lucas' approach and with the generalized WA [12]. WA reaches practically machine precision (13 significant digits) with 10 intervals (=partial integrals), while after 20 intervals, Lucas provides only 10 significant digits.…”

Section: Extension To Products Of Oscillating Functionsmentioning

confidence: 99%

“…Less recognized is the obvious fact that, once evaluated, the inner integral (32) will show some type of singularity if the domains P and Q touch at a point, share an edge or are identical [22]. In all these cases, DE should be a preferred choice to evaluate the outer integral (33).…”

Section: De and Multidimensional Singular Integralsmentioning

confidence: 99%

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Weighted Averages and Double Exponential Algorithms

Mosig

2013

IEICE Trans. Commun.

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SUMMARYThis paper reviews two simple numerical algorithms particularly useful in Computational ElectroMagnetics (CEM): the Weighted Averages (WA) algorithm and the Double Exponential (DE) quadrature. After a short historical introduction and an elementary description of the mathematical procedures underlying both techniques, they are applied to the evaluation of Sommerfeld integrals, where WA and DE combine together to provide a numerical tool of unprecedented quality. It is also shown that both algorithms have a much wider range of applications. A generalization of the WA algorithm, able to cope with integrands including products of Bessel and similar oscillatory functions, is described. Similarly, the original DE algorithm is adapted with exceptional results to the evaluation of the multidimensional singular integrals arising in the discretization of Integral-Equation based CEM formulations. The new possibilities of WA and DE algorithms are demonstrated through several practical numerical examples.

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Section: Extension To Products Of Oscillating Functionsmentioning

confidence: 99%

Section: Extension To Products Of Oscillating Functionsmentioning

confidence: 99%

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Weighted Averages and Double Exponential Algorithms

Mosig

2013

IEICE Trans. Commun.

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“…Using realistic values consistent with published data for benzene, soil and house, numerical values of concentrations throughout are obtained by numerical Laplace transform inversion using Talbot's method (Talbot, 1979). The evaluation of the double integrals entails the evaluation of infinite integrals containing products of Bessel functions which are performed using the method of Lucas (1995).…”

Section: Introductionmentioning

confidence: 99%

One- and Three-Dimensional Soil Transportation Models for Volatiles Migrating from Soils to House Interiors

Robinson

Turczynowicz

2005

Transp Porous Med

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Two models are presented for the transient migration of a volatile organic compound (VOC) from soil to the interor of a house with a crawl space. The migration in the house is taken as one-dimensional (1D) and coupled to a soil transportation model with diffusion, leaching and VOC degradation. The diffusion is either vertical, providing a 1D model, or three-dimensional (3D) axisymmetric, providing a 3D model. The initial subsurface VOC deposition is assumed to be a finite layer extending over the footprint of the house in the 1D case or as a cylinder of arbitrary radius in the 3D case. Using data for benzene as the VOC, comparisons are made for results from the two models for VOC concentrations in soil, crawl space and dwelling space as well as the cumulative dwelling space concentration used in human health assessment, for a wide range of soil parameters and subsurface sizes of the initial VOC cylinder. In all cases the values from the 1D transport model were slightly higher than those for the 3D model. The analysis uses Laplace transforms with numerical inversions. For the 3D soil transport model, different soil surface flux boundary conditions underneath and outside the house give rise to novel dual integral equations.

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“…in computing the filter loss coefficient in optical fibre technology [11], surface displacement in dynamic pavement testing [16], antenna theory [17], gravitational fields of astrophysical discs [5], crack problems in elasticity [22], particle motion in an unbounded rotating fluid [6,21], theoretical electromagnetics [8] or distortions of nearly circular lipid domains [20]. Techniques to compute this type of integrals are discussed in [2,14,22]. Analytic expressions for some special cases can be found in [19,26].…”

Section: Introductionmentioning

confidence: 99%

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

Deun

Cools

2006

Lecture Notes in Computer Science

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Abstract. 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.

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Evaluating infinite integrals involving products of Bessel functions of arbitrary order

Cited by 79 publications

References 8 publications

Weighted Averages and Double Exponential Algorithms

Weighted Averages and Double Exponential Algorithms

One- and Three-Dimensional Soil Transportation Models for Volatiles Migrating from Soils to House Interiors

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

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