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I Integration above or below axis

  1. Dec 17, 2017 #1
    If you are trying to find the integral of a function with a complex pole on the real axis (i.e. ##[-\infty,\infty]##) via analytic continuation, does it matter if you integrate along a contour either in the upper-half plane or lower-half plane? I was under the assumption that either approach was equivalent. For example, if you have a pole at ## a + ib##, where ##a## and ##b## are positive constants, and take your contour to be a semicircle with radius out to infinity, so that by Jordan's lemma that component of the integral goes to 0, then shouldn't either approach (i.e. top half or bottom half of a circle) give you the same answer? It seems that I am missing or incorrectly assuming something obvious here.

    (I've attached an image that attempts to depict what I tried to explain above.)

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  2. jcsd
  3. Dec 18, 2017 #2


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    The integral over your arcs doesn’t vanish with that function unless f(x) has some special properties (that will lead to additional poles).

    1/x (here: 1/r) is not sufficient as the length of the arcs scales with r and r/r=1 doesn’t converge to 0 for r to infinity.
  4. Dec 18, 2017 #3


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    To be more specific, Jordan’s lemma applies to functions that can be written on the form ##f(z) = e^{iaz} g(z)## with ##a## real and non zero. The sign of ##a## determines which half circle you close and the behaviour of ##g## determines whether or not the integral over that half circle vanishes.
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