A Numerical Integration Formula Based on the Bessel Functions

Abstract

In this paper, we discuss the properties of a quadrature formula with the zeros of the Bessel functions as nodes for integrals ${\int }_{-\infty }^{\infty }|x{|}^{2\nu +1}f(x)dx$, where $\nu$ is a real constant greater than $-1$ and $f(x)$ is a function analytic on the real axis $(-\infty ,+\infty )$. We show from theoretical error analysis that (i) the quadrature formula converges exponentially, (ii) it is as accurate as the trapezoidal formula over $(-\infty ,+\infty )$ and (iii) the accuracy of the quadrature formula doubles that of an interpolation formula with the same nodes. Numerical examples support the above theoretical results. We also apply the quadrature formula to the numerical integration of integral involving the Bessel function.