Gauss–Legendre quadrature for the evaluation of integrals involving the Hankel function

The boundary integral method for the two dimensional Helmholtz equation requires the approximate evaluation of the integral ∫-11g(x)H0(1)(λ(x-a)2+b2)dx, where g   is a polynomial. In particular, Gauss–Legendre quadrature is considered when the source point is close to the interval of integration; that is -1⩽a⩽1-1⩽a⩽1 and 0