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Residue & pole of a function

  1. Apr 16, 2006 #1


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    Here I must evaluate

    [tex]\frac{1}{2\pi i}\int_C \frac{f(\zeta)}{w(\zeta)}\frac{w(\zeta)-w(z)}{\zeta -z}d\zeta[/tex]

    where f is holomorphic on the hole complex plane, where w(z) is a polynomial of degree n with all of its zeroes distinct (i.e. all n have multiplicity 1), and where C is a closed curve containing all the zeroes of w in its interior.

    The solution manual says that the integral equals

    [tex]\sum_{j=1}^n Res(\frac{f(\zeta)}{w(\zeta)}\frac{w(\zeta)-w(z)}{\zeta -z}, \zeta_j)[/tex]

    where [itex]\zeta_j[/itex] are the n zeroes of w. But isn't [itex]\zeta = z[/itex] also a pole for the integrand?
  2. jcsd
  3. Apr 17, 2006 #2


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    If the contour excludes [itex]\zeta = z[/itex] then the pole will not contribute to the integral.
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