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Is it possible to calculate integral $$ \int\limits_0^{2\pi} |\sin t| dt \qquad (1) $$ by the complex function theory instruments, i.e. $$ \frac{1}{2\imath}\int\limits_\mathcal{C} \left|\frac{z^2-1}{z}\right|\frac{1}{z} dz, $$ 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 $$ P.V.\int\limits_0^{2\pi} \cot\frac{t-t'}{2}|\sin t| dt, \qquad (2) $$ or even $$ P.V.\int\limits_0^{2\pi} e^{\imath m(t-t')}\cot\frac{t-t'}{2}|\sin t| dt, \qquad (3) $$ 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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The absolute value function (and hence most compositions with it) is not analytical / holomorphic. So in general I would say you cannot use Cauchy (or anything else assuming analyticality) on something containing an absolute value. Of course, there are special cases, but $|\sin z|$ along a circle in the complex plane is not one of them. –  Arthur Sep 24 '11 at 16:05
    
What are some of these more complicated integrals that you talk about? –  mixedmath Sep 24 '11 at 16:19
    
@mixedmath please see explanation in updated question. –  V_V Sep 24 '11 at 16:52
    
I'm more concerned with the cotangents than with the absolute values myself... –  J. M. Sep 24 '11 at 18:34

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