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Math Help - Cauchy integral prob - not sure if i am interpreting it corrrectly

  1. #1
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    Talking Cauchy integral prob - not sure if i am interpreting it corrrectly

    Consider the integral ∮dz/(z-a) around |z| = 1 oriented counter-clockwise, where a is any constant such that |a| ≠ 1. Using Cauchy's Theorem and Cauchy's Integral Formula, evaluate this integral for the cases where |a| > 1 and |a| < 1.

    I used Cauchy's Integral Formula: f(a) = 1/2πi ∮f(z)dz/(z-a)

    since|a|>1 then we don't have to worry about z = a becasue |z| = 1

    so i rearranged...

    f(a)2πi = ∮f(z)dz/(z-a)
    = f(a)2πi = ∮dz/(z-a)
    = f(a)2πi = ∮z/(z-a)
    = f(a)2πi = ∮[z/(z-a)][1/(z - 0)]dz

    this then gives a new function g(z) = z/(z-a) which is analytic in the circle

    so now we are doing this for g(0)

    = g(0)2πi = ∮[z/(z-a)][1/(z - 0)]dz

    = 0/(0-a)

    =0


    for |a| < 1

    i just used Cauchy's Integral such that

    f(a)2πi = ∮f(z)dz/(z-a) where f(z) = 1 in the given example... so f(a) = 1

    so f(a)2πi

    = 2πi

    sooo... my question is; am I doing this correctly?
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  2. #2
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    Quote Originally Posted by douber View Post
    Consider the integral ∮dz/(z-a) around |z| = 1 oriented counter-clockwise, where a is any constant such that |a| ≠ 1. Using Cauchy's Theorem and Cauchy's Integral Formula, evaluate this integral for the cases where |a| > 1 and |a| < 1.

    I used Cauchy's Integral Formula: f(a) = 1/2πi ∮f(z)dz/(z-a)

    since|a|>1 then we don't have to worry about z = a becasue |z| = 1

    so i rearranged...

    f(a)2πi = ∮f(z)dz/(z-a)
    = f(a)2πi = ∮dz/(z-a)
    = f(a)2πi = ∮z/(z-a)
    = f(a)2πi = ∮[z/(z-a)][1/(z - 0)]dz

    this then gives a new function g(z) = z/(z-a) which is analytic in the circle

    so now we are doing this for g(0)

    = g(0)2πi = ∮[z/(z-a)][1/(z - 0)]dz

    = 0/(0-a)

    =0


    for |a| < 1

    i just used Cauchy's Integral such that

    f(a)2πi = ∮f(z)dz/(z-a) where f(z) = 1 in the given example... so f(a) = 1

    so f(a)2πi

    = 2πi

    sooo... my question is; am I doing this correctly?

    Yeppers. One thing only: when you got 0 you assumed, of course, |a| > 1, right?
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  3. #3
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    yep i was doing |a| > 1

    thanks for the input. i thought i was on the right track
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