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QUESTION


Find $\int_{\alpha}{\rm  Log}z dz$ where $\alpha$ is the contour
defined by $z=e^{\pi it}$ for $0\leq t\leq \frac{1}{4}$.



ANSWER


Put $z=e^{\pi it}$, so $dz=\pi ie^{\pi it}dt$, and Log$z=i\pi t$
and we get $\int_{\alpha}{\rm Log}z dz=-\pi^2\int_0^{1/4}te^{\pi
it}dt$ which we evaluate using integration by parts.




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