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The integral in question is

$\int_{_C} \frac{z}{cos(z)}\,dz,$ Where C is path $e^{jt},$ where $\ 0< t < 2\pi$

Since the pole of the function is +$\frac{\pi}{2}$ and -$\frac{\pi}{2}$, both of which are outside the path of integration, does that mean the integral equals to zero?

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    By Cauchy's, yes.2012-06-21
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    Of course, the integrand has more poles than that – but the rest of them are even further outside the path of integration, so you're safe from them.2012-06-21

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