Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
1  
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

1 Answer 1

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.

share|improve this answer
    
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. –  Joshua May 14 '12 at 2:11

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.