On the evaluation of correction terms in Gauss–Legendre quadrature

In the numerical integration of analytic functions, the singularities of the integrand affect the rate of convergence of the quadrature. This convergence can be improved significantly by adding the residue correction terms for the poles of the integrand. But this needs the evaluation of the basis function and its corresponding second kind function with complex arguments. We indicate a simple and accurate method to evaluate the correction term involving the basis and its second kind functions in the case of Gauss–Legendre quadrature. This approach does not call for the evaluation of the hypergeometric functions.