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Math Help - contour integrals

  1. #1
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    contour integrals

    Compute integral of:
    Integral over C of 1/z dz
    where C is the unit circle centered at some point z.
    |z.| >2
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  2. #2
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    well a unit circle in 1 not z= 2

    try putting z into polar form and then do a contour integral
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  3. #3
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    the center of the circle is at a point z, |z|>2 its just saying the unit circle is shifted from the origin..
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  4. #4
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    would it be written then |z-2|>2
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  5. #5
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    Quote Originally Posted by stumped765 View Post
    the center of the circle is at a point z, |z|>2 its just saying the unit circle is shifted from the origin..
    Then use something like z_0 to differentiate it from the variable z.

    The unit circle about z_0 with radius 1 is |z-z_0|= 1 and z= z_0+ e^{i\theta} with \thetaa going from 0 to 2\pi.

    Then dz= ie^{i\theta}d\theta \oint \frac{1}{z}dz= \int_0^{2\pi}\frac{ie^{i\theta}}{z_0+ e^{i\theta}}d\theta.

    However, you should know that \frac{1}{z} is analytic everywhere except where z= 0 which, with |z- z_0|> 2, means everywhere inside this contour.
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  6. #6
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    Quote Originally Posted by stumped765 View Post
    Compute integral of:
    Integral over C of 1/z dz
    where C is the unit circle centered at some point z.
    |z.| >2
    Hi. Here's something fun to check your knowledge about contour integrals: pin the center of the unit circle at the point 2i. Now, drop it straight through the singular point of 1/z at the origin until the center rests at -2i. Now, how does the value of the integral \mathop\oint\limits_{C} \frac{1}{z}dz change as the circle falls?
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