Numerical Quadrature Rules for Some Infinite Range Integrals

Abstract

Recently the present author has given a new approach to numerical quadrature and derived new numerical quadrature formulas for finite range integrals with algebraic and/or logarithmic endpoint singularities. In the present work this approach is used to derive new numerical quadrature formulas for integrals of the form $\smallint _0^\infty {x^\alpha }{e^{ - x}}f(x) dx$ and $\smallint _0^\infty {x^\alpha }{E_p}(x)f(x) dx$, where ${E_p}(x)$ is the exponential integral. It turns out the new rules are of interpolatory type, their abscissas are distinct and lie in the interval of integration and their weights, at least numerically, are positive. For fixed $\alpha$ the new integration rules have the same set of abscissas for all p. Finally, the new rules seem to be at least as efficient as the corresponding Gaussian quadrature formulas. As an extension of the above, numerical quadrature formulas for integrals of the form $\smallint _{ - \infty }^{ + \infty }|x{|^\beta }{e^{ - {x^2}}}f(x) dx$ too are considered.