Numerical evaluation of integrals containing a spherical Bessel function by product integration

1981 ◽  
Vol 22 (7) ◽  
pp. 1399-1413 ◽  
Author(s):  
D. R. Lehman ◽  
William C. Parke ◽  
L. C. Maximon
Keyword(s):  
Bessel Function ◽  
Numerical Evaluation ◽  
Spherical Bessel Function ◽  
Product Integration

Spherical Bessel function based deterministic beam‐forming for spherical‐surfaced apertures

2013 ◽  
Vol 49 (14) ◽  
pp. 863-865 ◽  
Author(s):  
K.W. Xu ◽  
H. Li ◽  
L. Chen ◽  
J.T. Huangfu ◽  
L.X. Ran
Keyword(s):  
Bessel Function ◽  
Beam Forming ◽  
Spherical Bessel Function

Tables for numerical integration at non-equidistant argument steps

1952 ◽  
Vol 48 (3) ◽  
pp. 436-442
Author(s):  
H. O. Hartley
Keyword(s):  
Numerical Integration ◽  
Numerical Evaluation ◽  
Definite Integral ◽  
Weighted Sum ◽  
Integration Formula ◽  
Product Integration ◽  
Equal Variance ◽  
Observational Errors ◽  
Image Position ◽  
Weight Coefficients

1. Published tables facilitating the numerical evaluation of the definite integralfrom given numerical valuesf(xi) of the integrand in the form of a weighted sumessentially deal with the following situations:(a) The argumentsxiare equidistant, i.e.xi=x0+iw;i=integer and the weight coefficientsciare tabulated for all argumentsxioccurring in the summation. Among these tables the most important are:(a1) panels in which the end-points of the integration a andbare two of thexi, and in particular(a11) panels confined to the range of the integration in which(a12) panels extending beyond the range of integration which usually have the same number of arguments,k, below a and beyondb, so that(a2) panels in which a and b are at distance ½wfrom one of thexi, and in particular(a21) panels confined to the range of integration in which(a22) panels extending beyond the range of integration which usually have the same number of (k) points belowaand beyondb, so that(b) Thenargumentsxiare not equidistant but are specially chosen so that for a givennthe sumSapproximates toIwith highest precision. Gauss's formula (see, for example, Milne-Thomson (4)) and various formulae of approximate product integration (see, for example, Beard (l)) are examples of these. In Chebyshev's integration formula (see, for example, Runge and König (5)) thexirequired for equal weights ctare used, resulting in an important formula when the ordinatesf(xi) are affeected by observational errors of equal variance.

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