Weighted quadrature formulas for semi-infinite range integrals

Abstract

Weighted quadrature formulas on the half line $(a,+\infty)$, $a>0$, for non-exponentially decreasing integrands are developed. Such $n$-point quadrature rules are exact for all functions of the form $x\mapsto x^{-2}P(x^{-1})$, where $P$ is an arbitrary algebraic polynomial of degree at most $2n-1$. In particular, quadrature formulas with respect to the weight function $x\mapsto w(x)=x^\beta\log^m x$ ($0\le \beta<1$, $m\in \mathbb{N}_0$) are considered and several numerical examples are included