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When finding a residue, how am I to know which coefficient to choose? For instance, if I have, let's say three poles, which coefficients of the Laurent series do I choose to calculate the residue? Does it just go by the number of poles? 3 poles, the first, second, and third coefficients of the series?

I know the Residue of f(z) @ z=z_o is the first coeffiient, b_1, of the Laurent series. But after that, things get fuzzy. It does not matter what the value of z_o is, it is always the first coefficient? And if there are several poles... then what?

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A residue is a local concept. There is one at each point. Compute a Laurent expansion at a particular point $z_0$ and the residue at that point is the coefficient of $1/(z-z_0)$. If the function is holomorphic at a point then the residue at that point is 0. A function can have different residues at different points. For instance, $1/(z^2-z)$ has residue $-1$ at 0 and residue 1 at 1. – KCd May 11 '12 at 23:31
up vote 3 down vote accepted

A meromorphic function may have many poles, but each pole has exactly one residue. The residue is, as you say, the $-1$ coefficient in the local Laurent series around the pole. It is not any of the other coefficients. So when you are doing a residue theorem computation and you are supposed to sum the residues of the function, you take the $-1$ coefficient of each of the poles, and add them together. Hope that helps.

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Thx! That's what I was lookin' for and am not sure why I didn't see it before - I've been able to get up through some basic improper integrals this weekend. – user5262 May 14 '12 at 2:11

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