TITLE: How to Integrate along a path
QUESTION [2 upvotes]: 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?

REPLY [1 votes]: Trying to save one more unanswered question from doom:
The poles of the function are in the points where 
$$\cos z=0\Longleftrightarrow z=\left(n-\frac{1}{2}\right)\pi\,,\,n\in\Bbb Z$$
Since the path is
$$C:=\{z\in\Bbb C\;:\; |z|=1\Longleftrightarrow z=e^{it}\,\,,\,0\leq t\leq 2\pi\}$$
the unit circle centered at the origin, and this path contains in its interior no poles of the function, we get by Cauchy's Integral Theorem that
$$\oint_C\frac{z}{\cos z}dz=0$$
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