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Homework Help: Integral calculations using Cauchy's Integral formula

  1. May 3, 2010 #1
    1. The problem statement, all variables and given/known data

    Calculate the integral

    Given [tex]\int_{C} \frac{e^z}{\pi i - 2z} dz = \int_{C} \frac{e^z}{z-\frac{\pi i}{2}} dz} [/tex]

    using Cauchy integral formula.

    2. Relevant equations

    What I know

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

    3. The attempt at a solution

    This in my little girly mind amounts to

    [tex]2\pi i f(\frac{\pi i}{2}) = \int_{C} \frac{e^z}{z-\frac{\pi}{2}i} dz \Rightarrow \int_{C} \frac{e^z}{z-\frac{\pi}{2}i} dz = 2 \pi \cdot (i) \cdot (i) = -2\pi[/tex]

    But people who are wiser than me says to me "Susanne your result is wrong!". Could someone please point out my mistake?

    thanks Susanne
    Last edited: May 3, 2010
  2. jcsd
  3. May 3, 2010 #2


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    Science Advisor
    Homework Helper

    e^z/(i*pi-2z) isn't equal to e^z/(z-i*pi/2). It's equal to (-1/2)*e^z/(z-i*pi/2). You can't just drop the (-1/2) constant.
  4. May 3, 2010 #3

    You are a genius my man. I simply couldn't see that :D
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