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Homework Help: How to calculate residium at infinity?

  1. Jan 14, 2010 #1
    i have such a function
    [tex]
    z^3 \sin \frac{1}{3}
    [/tex]

    i need to calculate its residium at z=infinity

    if i substitue infinity instead of a"" into the formal formula
    [tex]res(f(x),a)=\lim_{x->a}(f(x)(x-a))[/tex]

    i get infinity
    am i correct?
     
  2. jcsd
  3. Jan 14, 2010 #2
    Consider first a pole at some point z = p. You know that 2 pi i times the residue there is the value of the contour integral that encirles that pole anti-clockwise and no other poles. If you then perform the conformal transform u = 1/z in that contour integral, you get the contour integral of:

    -1/u^2 f(1/u)

    which encircles the corresponding pole in the u-plane at u = 1/p. Note that the contour integral is traversed anti-clockwise if the contour does not encircle the origin. So, the residue at z = p of f(z) is the same as the residue of -1/u^2 f(1/u) at u = 1/p. The residue at infinity is defined such that this result holds in the limit p to infinity. So, it defined as the residue at zero of -1/z^2 f(1/z).
     
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