Cauchy's integral theorem example help!!?

integral (cos z dz)/(z-π)^2 where circle is positively oriented centered z=3, radius r=1.

i think cauchy's integral theorem was used. but i cannot understand the result. how do i get there? should i use this theorem? thanks

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  • kb
    Lv 7
    8 years ago
    Favorite Answer

    Note that z = π is inside the circle C: |z - 3| = 1.

    So, we have by Generalized Cauchy Integral Formula (as cos z is analytic inside and on C)

    ∫c cos z dz / (z - π)^2

    = ∫c cos z dz / (z - π)^(1+1)

    = 2πi * (d/dz) cos z {at z = π}

    = 2πi * -sin z {at z = π}

    = 0.

    I hope this helps!

  • 4 years ago

    Integrand has a pole at 0 and, with given center and radius, the circle winds around the pole as quickly as. Cauchy's formula says which you get 2pi(i)cos(0)(winding), the suggested answer. Radius below 4 could supply answer of 0.

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