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I don't Understand Principle Values

  1. Sep 24, 2009 #1
    Say, you have a line that divides the complex plane into two parts, one contains the poles and the other one doesn't. You can draw two separate arcs to complete a closed contour integral, one way is to encircle the half plane with the poles the other is to encircle the half plane without the poles. If we encircle the half plane with the poles Supposedly the contour integral of this arc goes to zero as the radius approaches infinity. Why isn't this also the case for the arch which encircles the half plane without the poles?
     
  2. jcsd
  3. Sep 24, 2009 #2
    It depends on the function. Usually a meromorphic function does NOT go to zero in all directions. Only in some. You have to choose the contours to use to get good estimates.
     
  4. Sep 24, 2009 #3
    I figured out what I was missing. Specifically I was looking at inverse laplace transforms from a book in complex analysis I had for over 10 years. What may be missed by some readers without reading back some in the book is that the Jordan inequality.


    [tex] \int_0^\pi e^{-Rsin(\theta)}d\theta<\frac{\pi}{2}[/tex]

    depends on the value of theta lying between 0 and pi over 2. The importance of the range of theata is used in the derivation but not highlighted in the book, thus one is left to wonder why one half arc integrates to zero (in the limit as R approaches infinity) and the other does not. Also not mentioned was the fact the Cauchy Schwarz inequality was used but I guess it should be safe to assume by this point in the book the readers should know that inequality well.
     
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