Comment on: A Gaussian quadrature for the optimal evaluation of integrals involving Lorentzians over a semi-infinite interval

✍ Scribed by Herbert H.H. Homeier; E.Otto Steinborn

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 221 KB
• Volume: 99
• Category: Article
• ISSN: 0010-4655
• DOI: 10.1016/s0010-4655(96)00115-4

✦ Synopsis

Gauss quadrature rules corresponding to weight functions (1 + x2) -" on the interval (0,~x~) have been proposed R.E Sagar, V.H. Smith Jr. and A.M. Simas, Comput. Phys. Commun. 62 (1991) 16) for the evaluation of atomic momentum expectation values. In this comment it is shown that by using Gauss-Rational quadrature rules the results of Sagar et al. can be improved considerably for higher accuracy demands. In addition, it is pointed out that up to now there ~s no sufficient proof that their procedure is convergent. The usual proof for Gauss rules does not apply. The reason is that for weight functions of the above form a complete orthogonal system of polynomials is not available due to the divergence ,)f the higher moment integrals. Sagar, Smith, and Simas (SSS) proposed [ 1 ] to use Gaussian quadrature rules based on weight functions of :he form

for the evaluation of certain momentum space integrals over the interval (0, cx~). The integrals to be evaluated are moments of spherically averaged momentum densities of the form OO (xg}=4cr/xk+2p(x) dx, -2<k<4.

0 (2)

Note that these moments can also be denoted equally well as (pk/, corresponding to an integration variable p, in order to stress their relation to momentum space as done by SSS. As a benchmark example they use an