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Cauchy Integral Number 2 (Proof)

  1. Oct 17, 2007 #1
    1. The problem statement, all variables and given/known data
    If f(z) is analytic interior to and on a simple closed contour C, then all the derivatives [tex]f^k(z)[/tex] , k=1,2,3... exist in the domain D interior to C, and

    Prove for second derivative.

    3. The attempt at a solution
    For the first derivative, I would start with


    where R= [tex]\frac{h}{2i\pi}\int_{C}\frac{f(\zeta)d\zeta}{(\zeta-z)^2(\zeta-z-h)}[/tex]

    By the ML Theorem, we end up with the R term, and therefore the whole term going to zero, so that all that is left is:


    For the second derivative, I would start with

    Again, I get a term R that goes to zero, and am suppose to be left with


    Where does the two in the numerator come from? I'm confused by the calculations.
  2. jcsd
  3. Oct 17, 2007 #2


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    First, there was no "H" to disappear! You mean, of course, "h", but you need to be careful about that: in general "H" and "h" may be very different things.

    What do you get if you actually do the subtraction? What is

  4. Oct 17, 2007 #3
    Oh! Duh... you get


    And that h cancels with the 1/h on the outside.

    Does anyone see where the 2! in the numerator of
    Last edited: Oct 17, 2007
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