Computing Residues

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How might one go about computing the residue of $\frac {z^2 + 3z - 1}{z+2}$? I understand it has a pole at -2 and that we should then expand the numerator in powers of 2, but the book seems to do it by inspection. How does it look when done methodically?

EDIT: I should clarify - I understand I can just plug -2 into the numerator but I do not know why.

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1 Answer

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Here is how check the order of the pole and notice that for the case of a simple pole it equals the residue. Now, $z=-2$ is a simple pole then the residue is the coeffiecient of $a_{-1}$ and it is given by

$$ \lim_{z\to -2}(z+2)\frac {(z^2 + 3z - 1)}{(z+2)}=\dots\,. $$

Other related problems (I), (II).

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