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  1. #1
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    Cauchy's Integral Forumla

    Please could someone give me some insight into this question
    Question: Evaluate
    [Math]\oint[/tex](e^z)/(z^3+z) dz
    within a circle with radius 2


    Sorry i could get the latext to work.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Cauchy's Integral Forumla

    Quote Originally Posted by Downes View Post
    Question: Evaluate [Math]\oint[/tex](e^z)/(z^3+z) dz within a circle with radius 2
    A generic circle with radius 2?. A particular circle with radius 2?
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  3. #3
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    Re: Cauchy's Integral Forumla

    To use Cauchy's Integral Formula, as you title this thread, I would note that z^3+ z= z(z+ i)(z- i) then use use "partial fractions" to write the integrand as a sum of three fractions and then do three separate integrals, using Cauchy's Integral Formula on the three separate integrals.

    As FernandoRevilla implies, the answer depends upon the particular contour. In fact, because you have three different points in question, z= 0, z= i, and z= -i, and the value of each integral depends upon whether the point is or is not in the contour, there will be 2^3= 8 answers. For example, if the contour is such that NONE of the three points are in it, the integral in question is 0. If z= 0 is in the integral but not -i or i, you will get one answer, if z= i is in the integral but not 0 or -i, another, if both 0 and i are in the contour but not -i, yet another, etc. (Since the radius is given as 2, you cannot have all three points inside the contour so there are really at most 7 different answers.)
    Last edited by HallsofIvy; August 22nd 2011 at 10:36 AM.
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  4. #4
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    Re: Cauchy's Integral Forumla

    Thanks for the quick answer
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