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Complex integral question

  1. Jul 15, 2010 #1
    [tex]\int_{|z|=3}^{nothing}\frac{dz}{z^3(z^{10}-2)}\\[/tex]
    [tex]f=\frac{1}{z^3(z^{10}-2)}\\[/tex]
    [tex]f(\frac{1}{z})=\frac{1}{(\frac{1}{z})^3((\frac{1}{z})^{10}-2)}\frac{z^{13}}{1-2z^{10}}=\\[/tex]
    [tex]res(f,\infty)= res(\frac{1}{z^2}f(\frac{1}{z}),0)=\frac{1}{z^2}\sum_{n=0}^{\infty}(2z^{10})^n\\[/tex]
    [tex]res(f,\infty)= res(\frac{1}{z^2}f(\frac{1}{z}),0)=-[res(f,inside|z|=3)+res(f,outside|z|=3)][/tex]

    from the sum i get that there is no [tex]z^{-1}[/tex] member in the series
    so the coefficient of [tex]z^{-1}[/tex] is zero

    so the residiu of infinity is zero
    but still all of my singular points are |z|=3
    so the integral equals zero
    ??

    did i solved it correctly
    did i written every formula regarding the laws of residue correctly here
    ?
     
  2. jcsd
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