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The Cauchy-Goursat Theorem

  1. Mar 19, 2013 #1
    The Cauchy-Goursat Theorem states:

    Let ##f## be holomorphic in a simply connected domain D. If C is a simple closed contour that lies in D, then
    ##\int_C f(z) \mathrm{d}t = 0 ## ​

    Now if ##f## is holo just on ##|C| \bigcup \ int(C)## (i.e ##f## holo only on the contour and inside of it, if we take ##z_0 \in \mathbb{C}## can we deduce from the theorem that

    ## \int_C \frac{f'(z)}{z-z_0} \mathrm{d}t = \int_C \frac{f(z)}{(z-z_0)^2} \mathrm{d}t = 0 ## whether ##z_0 \in \left[\ |C| \bigcup \ int(C) \right]## or not

    Since both equal ##0##.

    Also where does ##z## must belong to for the theorem to apply?
     
  2. jcsd
  3. Mar 19, 2013 #2
    I think I got it. I cannot apply Cauchy inside because of the singularity problem.

    Now my problem is, even though we don't know if ##f## is holo outside by the givens, can we still apply Cauchy outside ##C## and claim that

    [tex]\int_c{f(z)} \mathrm{d}t = 0[/tex]​
     
    Last edited: Mar 19, 2013
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