Evaluation of Bessel function integrals with algebraic singularities (Q1184781)

The author derives a new method for the numerical evaluation of the integral \(\int^ 1_ 0(1-x)^ \alpha x^ \beta f(x)\) \(J_ \nu(ax)dx\). Here \(\alpha\), \(\beta\), \(\nu\) and \(a\) are given constants; \(J_ \nu\) is the Bessel function of the first kind and order \(\nu\); \(f\) is a sufficiently smooth function so that it can be expanded into a series of the shifted Jacobi polynomials. The proposed method is based on the series expansion for \(J_ \nu(ax)\).