Modified quadrature formula for integrand with nearby poles

The conventional trapezoidal approximation for the numerical evaluation of the integral formula for the Dirichlet problem inside the unit disc becomes highly inaccurate when the point of evaluation is approaching the boundary of the circular domain. This is due to the presence of two nearby poles of the integrand function near the interval of integration. A modified quadrature formula is derived using the generalized Residue Theorem which provides much accurate numerical approximation of the integral formula compared with the trapezoidal approximation. Error estimates of the proposed numerical quadrature are also presented.