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2017

Indefinite Integrals of Spherical Bessel Functions

Bloomfield, Jolyon K., Face, Stephen H. P., Moss, Zander

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Highly oscillatory integrals, such as those involving Bessel functions, are best evaluated analytically as much as possible, as numerical errors can be difficult to control.

We investigate indefinite integrals involving monomials in $x$ multiplying one or two spherical Bessel functions of the first kind $j_l(x)$ with integer order $l$.
Closed-form solutions are presented where possible, and recursion relations are developed that are guaranteed to reduce all integrals in this class to closed-form solutions.
These results allow for definite integrals over spherical Bessel functions to be computed quickly and accurately.

Indefinite Integrals of Spherical Bessel Functions · Around

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R. Mehrem, J. T. Londergan, and M. H. Macfarlane, “Analytic expressions for integrals of products of spherical bessel functions,” Journal of Physics A: Mathematical and General
1991
Earlier work this paper cites.
L. C. Maximon, “On the evaluation of the integral over the product of two spherical bessel functions,” Journal of Mathematical Physics
1991
Earlier work this paper cites.
Elsevier, Amsterdam, Boston, Paris, et al., 2007
I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of integrals, series, and products · 2007
Earlier work this paper cites.

Similar

http://functions.wolfram.com/Bessel-TypeFunctions/SphericalBesselJ/21/01/
Wolfram Research, “Spherical bessel j integrals.” Website, 2007 · 2007
Cited alongside, same era.
http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/21/01/
Wolfram Research, “Bessel j integrals.” Website, 2007 · 2007
Cited alongside, same era.
R. Mehrem, “The Plane Wave Expansion, Infinite Integrals and Identities involving Spherical Bessel Functions,” Submitted to: Appl. Math. Comput
Original
2009
Cited alongside, same era.
W. Rosenheinrich, “Tables of some indefinite integrals of bessel functions,”. http://www.eah-jena.de/rsh/Forschung/Stoer/besint.pdf
Cited in the paper.

Then

V. I. Fabrikant, “Elementary exact evaluation of infinite integrals of the product of several spherical bessel functions, power and exponential,” Quart. Appl. Math
2013
Later among the works it cites.
http://dlmf.nist.gov/ . F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds
“NIST Digital Library of Mathematical Functions.” http://dlmf.nist.gov/, Release 1.0.14 of 2016-12-21 · 2016
Later among the works it cites.

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