Well, if you pick your semicircle so that the center of the semicircle is always 1/2 of a radius to the left of the singular point, for example, then shrink the circle while keeping this constraint, it will, indeed, converge to the singular point, but the left bound of the circle is 3 times as far away from the singular point as the right. I'm fairly sure that I worked an example (long ago, when I was taking a class in Complex Analysis), and the answer was different than the regular PV result. The "purpose" is to illustrate that it matters how one constructs the bypass of the singular point. I admit, however, that I am unfamiliar with the definition of legitimate contour. Bill C. ------------------------------------------------------------------------------ Quoting "Cordwell, William R" <wrcordw@sandia.gov>:

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.

Not really. When the singularity is a point, it should be surrounded by a small circle centered on the point. Otherwise the limiting value as the excision is shrunk will not work out, although it would likely be possible to concoct wierd limits. To what purpose? Other than to make clever trickery? Then again, maybe there is a legitimate contour in which the singularity sits at the tip of a corner that is only part of 360 or 180 degrees. -hvm