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# 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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- kbLv 78 years agoFavorite 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!

- demboskyLv 44 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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