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. Referring to the original question, I seem to have forgotten how the Fundamental Theorem of the Calculus would relate to a Lesbegue Integral. For a Riemann Integral, the variations that worried the student could probably be avoided by taking any partition of the interval of integration, and refining it in ways depending on the size of the maximum subinterval length and examining the neighborhood of zero length. As I recall, upper and lower Darboux sums confine the possible fluctuations within subintervals. Coinciding, disparities in the sample value for the interval become inconsequential, thereby eliminating another doubt. Some degree of continuity is implied, which seems to be why other kinds of integrals were invented. -hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos