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Let $C$ be $x^2+y^2=9$, oriented counterclockwise.

Find $$\int_{C}\frac{e^{\pi z}}{\frac{(z-4i)^2}{z-i}}dz$$

It is easy to find the parameterization of $C$. However, when it comes to the integral, I don't know how to deal with $z$.

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It appears you want to find $\int_C\frac{(z-i)e^{\pi z}}{(z-4i)^2}dz$. Check the integrand is analytic on and within $C$. – Host-website-on-iPage Oct 28 '12 at 8:48
up vote 1 down vote accepted

Check out Cauchy's differentiation formula and rearrange!

So here we have $\int_C\frac{(z-i)e^{\pi z}}{(z-4i)^2}dz$ so we differentiate $e^{\pi z}$ once and multiply by ${\pi i}$ $\left(=\frac{2\pi i}{2!}\right)$.

We do not need to worry about the pole at $4i$ in this case since this does not lie within $C$.

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