Error expansion of piecewise constant interpolation rule for certain two-dimensional Cauchy principal value integrals

The classical composite rectangle (midpoint) rule for the computation of two dimensional singular integrals is discussed, with the error functional of rectangle rule for computing two dimensional singular integrals, the local coordinate of certain point and the convergence results O(h2) are obtained. When the local coordinate is coincided with certain priori known coordinates, we get the convergence rate the same as the Riemann integral at certain point. At last, numerical examples are presented to illustrate our theoretical analysis which agree with it very well.