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Residue Theorem

  1. Mar 3, 2009 #1
    1. The problem statement, all variables and given/known data

    If f1 and f2 have residues r1 and r2 at z0. show that the residue of f1+ f2 is r1 + r2

    3. The attempt at a solution

    Res(f1, z0) = limz-->z0 (z-z0)f1(z) = r1
    Res(f2, z0) = limz-->z0 (z-z0)f2(z) = r2

    now calculate Res(f1+f2, z0)

    =

    limz-->z0 (z-z0)(f1(z)+ f2(z))
    = limz-->z0 (z-z0)(f1(z))+(z-z0)(f2(z) = r1+ r2

    Is it really this easy? i must be doing something wrong
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 3, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Pretty much, yeah. "Residue" is only defined for poles and a function has a pole at [itex]z_0[/itex] if and only if it can be expanded in a power series (a Laurent series) with a finite number of negative exponents. In that case the residue is the coefficient of z-1 (thus that limit formula you use). Adding the two functions, you can add the Laurent series term by term: [itex]a_1z^{-1}+ a_2z^{-2}= (a_1+ a_2)z^{-1}[/itex]. The residues add.
     
  4. Mar 3, 2009 #3
    Thanks halls.
     
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