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Let $\gamma$ be the circle in the complex plane given by $|z-i| = 1$, with counterclockwise orientation. Evaluate $$\int \frac{|z|^2}{(z-i)^2} dz$$

The function $|z|^2$ is not analytic. How would I set up the limits for this integral, is it just 0 to $2\pi$ because its a circle?

HokieFan7
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  • How do you want to parametrize the circle? This must be answered before you can even ask about the limits. – Brian Moehring May 03 '20 at 07:10
  • It's $$\int_0^{2\pi}\frac{|i+re^{it}|^2}{r^2e^{2rit}}ire^{it},dt$$ using the obvious parametrisation. – Angina Seng May 03 '20 at 07:11
  • I know for a circle its x(t) = rcost and y(t) = rsint, but I'm not sure how to apply that for this case. Could you explain how you got that integral? – HokieFan7 May 03 '20 at 07:13
  • Only a circle centered at the origin can be parametrized by $x = r\cos t, y = r\sin t$. This circle is centered at $i\ (x = 0, y = 1)$, not $0$. – Paul Sinclair May 03 '20 at 15:42

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