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Math Help - Complex Analysis: Cauchy's Theorem

  1. #1
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    Complex Analysis: Cauchy's Theorem

    As I am studying for an exam I am trying to wrap my head around the concepts I learned. I want to make sure I fully understand the concepts before the exam in 1.5 weeks.

    Cauchy's Theorem
    If u and v satisfy the Cauchy-Riemann equations inside and on the simple closed contour C, then the integral of f(z)=0

    Now for example, if we have f(z)=1/(z+20) and our closed contour is a circle around the origin with radius=1. If I am understanding this correctly, we can say that the integral is equal to 0 since the 'bad point' of z=-20 is outside of the circle correct meaning that f is differentiable in and on |z|=1.

    Does this sound correct?
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  2. #2
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    Quote Originally Posted by jokusan13 View Post
    As I am studying for an exam I am trying to wrap my head around the concepts I learned. I want to make sure I fully understand the concepts before the exam in 1.5 weeks.

    Cauchy's Theorem
    If u and v satisfy the Cauchy-Riemann equations inside and on the simple closed contour C, then the integral of f(z)=0

    Now for example, if we have f(z)=1/(z+20) and our closed contour is a circle around the origin with radius=1. If I am understanding this correctly, we can say that the integral is equal to 0 since the 'bad point' of z=-20 is outside of the circle correct meaning that f is differentiable in and on |z|=1.

    Does this sound correct?
    Yes, that is correct. The, around a closed path, of any function that is analytic inside and on the path is 0.
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