I am having trouble understanding how to calculate residues.

I have the function

[TEX]f(z)=\frac {z^a}{1+z^2}[/TEX] and am supposed to find

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

[TEX]\lim_{z\to+i}\frac{{z^a}{(z-i)}}{1+z^2}[/TEX] and [TEX]\lim_{z\to-i}\frac{{z^a}{(z+i)}}{1+z^2} [/TEX]

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

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