Quadrature formulas for infinite integrals

Abstract

have become increasingly important. The only quadrature generally available for the case b = - a = oo is the Hermite-Gauss formula although the LaguerreGauss formula can also be used if f(x) is an even function of x. The latter would, however, require computation of twice the number of ordinates for a corresponding degree of precision and would therefore rarely be preferred. In either case the integrand is supposed to behave like the product of an exponential function and a polynomial. For purely algebraic integrands it would appear to be more appropriate to use a quadrature based on an algebraic weight function even though the degree of the polynomial approximation to f(x) is limited. In this paper, formulas of type (1) are derived with weight function w(x) = (1 + x2) -k- for the range b = -a = cc. In a modified form they are shown to be superior to the Hermite-Gauss and Laguerre-Gauss quadratures for a particular class of statistical integrals.