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Math Help - Calculating a residue

  1. #1
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    Calculating a residue

    does anyone know how to work out the residue of this at 0

    i already have that
    Res(f,0)= .5 lim (2*pi^3*z*cot(pi*z) -2*pi^2)cosec^2(pi*z)
    (as z tends to 0)

    But i cant work out the limit, seeing as everytime i work it out i get different answers, i was told to use taylor series though
    Last edited by mr fantastic; January 5th 2009 at 12:27 PM.
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  2. #2
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    Quote Originally Posted by pols89 View Post
    does anyone know how to work out the residue of this at 0

    i already have that
    Res(f,0)= .5 lim (2*pi^3*z*cot(pi*z) -2*pi^2)cosec^2(pi*z)
    (as z tends to 0)
    Write cot(πz) as cos(πz)/sin(πz) and cosec(πz) as 1/sin(πz). Then you get \lim_{z\to0}\frac{\pi^2(\pi z\cos(\pi z) - \sin(\pi z))}{\sin^3(\pi z)}. Now use the power series expansions of sin and cos to find the lowest power of z with nonzero coefficient. I make the limit -\pi^2/3.
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