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Evaluating Integrals Using Cauchy's Formula/Theorem

  1. Mar 4, 2008 #1
    [SOLVED] Evaluating Integrals Using Cauchy's Formula/Theorem

    1. The problem statement, all variables and given/known data
    Evaluate the given integral using Cauchy's Formula or Theorem.

    [tex]\int_{|z+1|=2} \frac{z^2}{4 - z^2} \, dz[/tex]


    2. Relevant equations
    Cauchy's Theorem. Suppose that f is analytic on a domain D. Let [itex]\gamma[/itex] be a piecewise smooth simple closed curve in D whose inside [itex]\Omega[/itex] also lies in D. Then

    [tex]\int_\gamma f(z) \, dz = 0[/tex]

    Cauchy's Formula. Suppose that f is analytic on a domain D and [itex]\gamma[/itex] is a piecewise smooth, positively oriented simple closed curve D whose inside [itex]\Omega[/itex] also lies in D. Then

    [tex]f(z) = \frac{1}{2\pi i} \int_\gamma \frac{f(\zeta)}{\zeta - z} \, d\zeta[/tex]

    for all [itex]z \in \Omega[/itex].


    3. The attempt at a solution
    For z equal to 2 and -2, the integrand is not defined and hence is not analytic in any domain that includes these points. The problem is that the curve |z + 1| = 2 contains the point -2 inside it so I can't apply either Cauchy's Theorem or Formula. What can I do?
     
  2. jcsd
  3. Mar 4, 2008 #2
    hint

    z^2/(4 - z^2) = z^2/((2 - z)(2 + z)) = (z^2/(z-2))/(z+2) = g(z)/(z - (-2)), where g(z) = z^2/(z-2) is analytic inside and on your circle
     
  4. Mar 4, 2008 #3
    Ah! Thanks for the tip. That really helps.
     
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