The Principal Value of a complex integral also depends on how the semi-circle (or half-square, or half-rectangle) is drawn to miss the singularity. That is, the limit can still approach the singular point from both sides, but unevenly, and this can give a different answer from the case where the point is approached evenly from both sides. Bill C. ---------------------------------------- One might worry that using a partition in which some intervals are much wider than others would mess up the answer, but this isn't the case, in the limit where the mesh goes to zero. For a situation in which Stephen's qualms *do* apply, see the "cylinder area paradox": one can define a sequence of polyhedral approximations to the cylinder for which the surface areas of the approximations do not converge on the true surface area of the cylinder, on account of the triangular faces of the polyhedron getting too long and skinny in shape, even though they're shrinking away to zero diameter. (Anyone know a good web reference for this? Freida Zames' article "Surface Area and the Cylinder Area Paradox" is only available via JSTOR, as far as I can tell.) Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun