1. Residue Theory

I am having trouble understanding how to calculate residues.

I have the function

$f(z)=\frac {z^a}{1+z^2}$ and am supposed to find

$\int^\infty_0 f(z)\,dz$ using the keyhole contour. So I found singularities at +i and - i and am trying to find the residues there using

$\lim_{z\to+i}\frac{{z^a}{(z-i)}}{1+z^2}$ and $\lim_{z\to-i}\frac{{z^a}{(z+i)}}{1+z^2}$

Is ths the right way to find residues? What are these limits?

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