Evaluation of Bessel function integrals with algebraic singularities

Abstract

A new method for the numerical evaluation of ∫ 1 0(1-x) αx βJ v(ax)dx is proposed, where, α, β, ν, a are given constants, and Jν is the Bessel function of the first kind and of order ν. It is assumed that f is a smooth function. The method is based on approximation of the function f by a polynomial given in the form of a linear combination of the shifted Jacobi polynomials R( α,β+ν) k , and on the use of Chebyshev series expansion for the function J ν ( ax).