A Cauchy type singular integral equation along a finite real interval and with a weight function with complex singularities at the end-points of the integration interval can be numerically solved by reduction to a system of linear equations, by using an appropriate numerical integration rule associated with the Jacobi polynomials, in exactly the same way used for the case of real singularities. For the numerical solution of such an equation arising in plane elasticity crack problems and the evaluation of stress intensity factors at crack tips, the Lobatto-Jacobi numerical integration rule is the most appropriate.