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Math Help - Cauchy's integral formula help

  1. #1
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    Cauchy's integral formula help

    Q: Using Cauchy’s integral formula, compute the integral of g(z) over the circle of radius 3 centred at the origin, the contour integral being taken counterclockwise.

    g(z) =z^3 + z^2 − 5/z − 2

    i cant wrap my head around this formula or how to even begin solving it. all other examples had two 'poles' rather tha just (z-2)

    any help will be much appreciated.
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  2. #2
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    Can you set up the integral?
    What does Cauchy's integral formula tell you?

    By the way, it's not clear if you meant z^3+z^2-5/z -2 or z^3+z^2-\frac{5}{z -2}.
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  3. #3
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    sorry im not sure what you mean by set up.
    to be clear its (z^3+z^2-5)/(z-2)
    I know i have to use the denominator to find the singularities of g(z) but i do not know how to plug in the equation into the formula.
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by EoinCahill View Post
    sorry im not sure what you mean by set up.
    to be clear its (z^3+z^2-5)/(z-2)
    I know i have to use the denominator to find the singularities of g(z) but i do not know how to plug in the equation into the formula.
    That's a long shot from what you had written. You do know that a+b \times c \neq (a+b) \times c, right? Parentheses are not a luxury.

    It's clear that the only singularity is at z=2.
    Now can you write g(z)dz in the form \frac{f(z)}{z-2}dz, where f is holomorphic inside the contour? What does Cauchy's formula tell you about the integral of such a form along the contour?
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  5. #5
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    It's clear that the only singularity is at z=2.
    yes i thought that myself.
    Now can you write in the form , where is holomorphic inside the contour?
    so f(z) = z^3+z^2-5??

    that the integral is equal to 2.pi.i??
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  6. #6
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    Quote Originally Posted by EoinCahill View Post
    yes i thought that myself.


    so f(z) = z^3+z^2-5??

    that the integral is equal to 2.pi.i??
    Yes. No.

    You need to review Cauchy's Integral formula (see post #4 here for example (in your question n = 1): http://www.mathhelpforum.com/math-he...uchys-thm.html)
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  7. #7
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    ok this is what i have so far.

    f(z)= ( z^3+z^2-5 dz
    ------) z-2

    is this right???
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  8. #8
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    Quote Originally Posted by EoinCahill View Post
    ok this is what i have so far.

    f(z)= ( z^3+z^2-5 dz
    ------) z-2

    is this right???
    I said in my earlier reply that yes your f(z) was correct and no your answer was not correct. Where has the above come from - it is nothing more than a re-statement of the question! Did you read the post I refered you to? You need to compare the given integral to Cauchy's Integral Formula and:

    1. Identify f(z).

    2. Identify \alpha.

    Then you need to:

    3. Evaluate f(\alpha).

    4. Write down the answer.

    I just don't see where you can be stuck .... What part of the above can you not do?
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