Two formulae with nodes related to zeros of Bessel functions for semi-infinite integrals: extending Gauss–Jacobi-type rules

Abstract

For approximating integrals $$\varvec{\int _0^{\infty }\!\!} \varvec{x}^{\varvec{\alpha }}\varvec{f(x)dx}$$ ( $$\varvec{\alpha >-1}$$ ) over a semi-infinite interval $$\varvec{[0,\infty )}$$ with a given function $$\varvec{f(x)}$$ , two formulae, one of them new and another associated with an existing formula, are presented. They are constructed in a limiting process to a semi-infinite interval $$\varvec{[0,\infty ]}$$ with a linear transformation for a well-known approximation method, the Gauss–Jacobi (GJ) rule and its family rules: the Gauss–Jacobi–Radau (GR) and Gauss–Jacobi–Lobatto (GL) rules on a finite interval. This procedure was used in constructing our previous limit Clenshaw–Curtis-type formulae. The limit GJ formula (LGJ) constructed in this way uses as nodes zeros of the Bessel function $$\varvec{\varvec{J_{\alpha }(x)}}$$ squared after multiplied by a positive constant a and the limit GR (LGR) (and limit GL (LGL)) formula those with zeros of $$\varvec{J_{\alpha +1}(x)}$$ . The LGJ formula is also shown to be derived from the formula developed by Frappier and Olivier (Math. Comp. 60:303–316, 1993) for an integral on $$\varvec{[0,\infty )}$$ . The LGR and LGL formulae give the same and new formula. We show that for a function f(z) analytic on a domain in the complex plane z and satisfying some appropriate conditions, there exists a constant $${d>0}$$ such that the errors of both formulae are $$\varvec{O(e^{-2d/a})}$$ as $$\varvec{a\rightarrow +0}$$ . The average of the LGJ and LGR formulae gives smaller quadrature errors than each formula. Numerical examples confirm these behaviors and show that the LGJ and LGR formulae give asymptotically the same quadrature errors of opposite sign. Consequently, the LGR formula behaves like an anti-LGJ formula in the same way as the Lobatto rule for integrals on $$\varvec{[-1,1]}$$ behaving like the anti-Gauss rule.