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Is it possible to calculate integral $ \int\limits_0^{2\pi} |\sin t| \mathrm dt \qquad (1) $ by the complex function theory instruments, i.e. $ \frac{1}{2\imath}\int\limits_\mathcal{C} \left|\frac{z^2-1}{z}\right|\frac{1}{z} \mathrm dz, $ where $z=e^{\imath t}$ and $\mathcal{C}$ is a unit circle in complex plane.

I understand that we can take integral (1) using much simpler approach but actually I need to calculate more complicated integrals of that type (including $|\sin t|$), like, at least $ P.V.\int\limits_0^{2\pi} \cot\frac{t-t'}{2}|\sin t| \mathrm dt, \qquad (2) $ or even $ P.V.\int\limits_0^{2\pi} e^{\imath m(t-t')}\cot\frac{t-t'}{2}|\sin t| \mathrm dt, \qquad (3) $ where $t'\in[0,2\pi]$, $m\in \!I\!\! N$.

Previously, until $|\sin t|$ appeared I used complex function theory well for that. I know that it is possible to calculate integral (2) by splitting integral over $[0,\pi]$ and $[\pi,2\pi]$ and opening modulus and while this taking into account Cauchy principal value integral definition. But it does not looks so nice as complex function theory.

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    I'm more concerned with the cotangents than with the absolute values myself...2011-09-24

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