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 ∫ - 1 1 g ( x ) H 0 ( 1 ) ( λ ( x - a ) 2 + b 2 ) d x , 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 and 0 < b ⪡ 1 so that the integral is nearly weakly singular. It is shown that the real and imaginary parts of the integral must be considered separately. The sinh transformation can be used to improve the truncation error of the imaginary part, but must not be used for the real part. An asymptotic error analysis is given.