On the Integral of the Product of Three Bessel Functions over an Infinite Domain
Author
S. K. H. Auluck
Title
On the Integral of the Product of Three Bessel Functions over an Infinite Domain
Description
On the Integral of the Product of Three Bessel Functions over an Infinite Domain
Category
Academic Articles & Supplements
Keywords
URL
http://www.notebookarchive.org/2018-07-6xhd2x0/
DOI
https://notebookarchive.org/2018-07-6xhd2x0
Date Added
2018-07-15
Date Last Modified
2018-07-15
File Size
6.34 megabytes
Supplements
Rights
Redistribution rights reserved
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The Mathematica Journal
On the Integral of the Product of Three Bessel Functions over an Infinite Domain
On the Integral of the Product of Three Bessel Functions over an Infinite Domain
Fourier-Space Representation of Nonlinear Dynamics of Continuous Media in Cylindrical Geometry
S. K. H. Auluck
Fourier-space representation of the partial differential equations describing nonlinear dynamics of continuous media in cylindrical geometry can be achieved using Chandrasekhar–Kendall (C–K) functions defined over infinite domain as an orthogonal basis for solenoidal vector fields and their generating function and its gradient as orthogonal bases for scalar and irrotational vector fields, respectively. All differential and integral operations involved in translating the partial differential equations into transform space are then carried out on the basis functions, leaving a set of time evolution equations, which describe the rate of change of the spectral coefficient of an evolving mode in terms of an aggregate effect of pairs of interacting modes computed as an integral over a product of spectral coefficients of two physical quantities along with a kernel, which involves the following integral: ,involving the product of three Bessel functions of the first kind of integer order. This article looks at this integral’s properties using a semi-empirical approach supported by numerical experiments. It is shown that this integral has well-characterized singular behavior. Significant reduction in computational complexity is possible using the proposed empirical approximation to this integral.
((m,γ),(m',γ'),(m'',γ''))=rdr(γr)(γ'r)(γ''r)
∞
∫
0
J
m
J
m'
J
m''
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Introduction
Introduction
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Properties of the Triple-Bessel Integral
Properties of the Triple-Bessel Integral
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Approximation to the Triple-Bessel Integral
Approximation to the Triple-Bessel Integral
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Numerical Exploration of the Triple-Bessel Integral
Numerical Exploration of the Triple-Bessel Integral
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Proposed Approximate Formula for the Triple-Bessel Integral
Proposed Approximate Formula for the Triple-Bessel Integral
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Discussion
Discussion
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Summary and Conclusion
Summary and Conclusion
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Appendix 1
Appendix 1
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Appendix 2
Appendix 2
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Appendix 3
Appendix 3
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Appendix 4
Appendix 4
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Appendix 5
Appendix 5
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Appendix 6
Appendix 6
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Acknowledgments
Acknowledgments
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References
References
About the Author
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Cite this as: S. K. H. Auluck, "On the Integral of the Product of Three Bessel Functions over an Infinite Domain" from the Notebook Archive (2010), https://notebookarchive.org/2018-07-6xhd2x0
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