On the evaluation of the integral over the product of two spherical Bessel functions

Journal Article · Fri Mar 01 04:00:00 EST 1991 · Journal of Mathematical Physics (New York); (USA)

DOI:https://doi.org/10.1063/1.529405· OSTI ID:5694855

Maximon, L C ^[1]

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1. Department of Physics, The George Washington University, Washington, DC 20052 (US)

The integral {ital I}{sub {ital l},{ital l}{prime}}({ital k},{ital k}{prime})={integral}{sup {infinity}}{sub 0}{ital j}{sub {ital l}} ({ital kr}){ital j}{sub {ital l}{prime}}({ital k}{prime}{ital r}){ital r} {sup 2} {ital dr}, in which the spherical Bessel functions {ital j}{sub {ital l}}({ital kr}) are the radial eigenfunctions of the three-dimensional wave equation in spherical coordinates, is evaluated in terms of distributions, in particular, step functions and delta functions. It will be shown that the behavior of {ital I}{sub {ital l},{ital l}{prime}} is very different in the cases {ital l}{minus}{ital l}{prime} even (0, {plus minus}2, {plus minus}4, ...) and {ital l}{minus}{ital l}{prime} odd ({plus minus}1, {plus minus}3, ...). For {ital l}{minus}{ital l}{prime} even it is expressed in terms of the delta function, step functions, and Legendre polynomials. For {ital l}{minus}{ital l}{prime} odd it is expressed in terms of Legendre functions of the second kind and step functions; no delta functions appear.

• DOE Contract Number: FG05-86ER40270
• OSTI ID: 5694855
• Journal Information: Journal of Mathematical Physics (New York); (USA), Journal Name: Journal of Mathematical Physics (New York); (USA) Vol. 32:3; ISSN JMAPA; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BESSEL FUNCTIONS
COORDINATES
DELTA FUNCTION
DIFFERENTIAL EQUATIONS
DISTRIBUTION
EIGENFUNCTIONS
EQUATIONS
FUNCTIONS
INTEGRALS
LEGENDRE POLYNOMIALS
PARTIAL DIFFERENTIAL EQUATIONS
POLYNOMIALS
THREE-DIMENSIONAL CALCULATIONS
WAVE EQUATIONS