Complex analysis: laurent, residues

  • #1

Main Question or Discussion Point

This is addressed to people who know complex analysis (hope this is the right section). Here's the Laurent theorem from my book for my later reference: Suppose a function f is analytic throughout an annular domain R1<|z-z0|<R2, centered at z0, and let C denote any positively oriented simple closed contour around z0 and lying in that domain. Then, at each point in the domain, f(z) has the series representation f(z)=sum (n=0 to inf) of an(z-z0) + sum(n=1 to inf) bn/(z-z0)^n, where an=(1/ 2pi*i) integral over C of f(z)dz/(z-z0)^(n+1) and bn=(1/ 2pi i) integral over C of f(z)dz/(z-z0)^(-n+1).

So say there is a finite number of singularities of f, so they're all in a circle of some radius R. It seems to me you could select any point p in the complex plane, and then make an annulus around that point such that its inner perimeter encloses the singularities. The theorem seems to say that 2pi i *Res(f,p) is equal to the integral of a closed contour around the singularities. I know this is wrong b/c it doesn't agree with what I've seen with the Cauchy Residue Thm, where I calculated that integral by summing residues at singularities. Where is the disconnect?
 

Answers and Replies

  • #2
392
0
Good question.

b1 is not necessarily Res(f,z0) if the Laurent expansion is for R1<|z-z0|<R2 with R1>0.

If the Laurent expansion is for 0<|z-z0|<R2, then b1=Res(f,z0).
 
  • #3
1,707
5
i don't understand what billy bob said and i don't understand what you said.

residue theorem says the integral over a closed contour of f(z)/z-z0= res(f,z0) where the residue is the coefficient of the 1/z term in the laurent expansion. a corollary of cauchy goursat theorem states that if a function is analytic in some domain and contour encloses another contour the then integrals around those contours are equal. extending that a little and you get that the integral around a contour containing other contours equals the sum of integrals around each contour. this is why the integral of a function with several singularities around a contour equals the sum of the residues.

note that the laurent series is different in the neighborhood around each singularity.
 

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