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Find the residues of the function

  1. Sep 4, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the residues of the function

    2. Relevant equations

    f(z)=[itex]\frac{1}{sin(z)}[/itex] at z=0, [itex]\frac{∏}{2}[/itex], ∏

    3. The attempt at a solution
    Since the function has a simple pole at z=0

    I used: Res(f,0)=lim[itex]_{z->0}[/itex](z-0)[itex]\cdot[/itex][itex]\frac{1}{sin(z)}[/itex]=1. This means the residue of the function at z=0 is 1.

    And Res(f,[itex]\frac{∏}{2}[/itex])=lim[itex]_{z->∏/2}[/itex](z-[itex]\frac{∏}{2}[/itex])[itex]\cdot[/itex][itex]\frac{1}{sin(z)}[/itex]=0. This means the residue of the function at z=[itex]\frac{∏}{2}[/itex] is 0.

    However I think this method can not be applied on solving z=∏. How can I work it out?
     
    Last edited: Sep 4, 2012
  2. jcsd
  3. Sep 4, 2012 #2

    vela

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    Try expanding sin z in a series about ##z=\pi##. Or, better yet, use ##\sin z = \sin[(z-\pi)+\pi]##. The basic idea is to get everything in terms of ##z-\pi##.
     
  4. Sep 4, 2012 #3
    If I turn everything into z-∏, do we need to use the same method as I used to work out z=0 and z=[itex]\frac{∏}{2}[/itex]?
    But how? Turn the limit to z-∏→0?
    Because I used computer to work out the answer of z=∏ which is -1. But I still cannot get this answer by myself.
     
  5. Sep 4, 2012 #4

    vela

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    You find the residue by evaluating the limit
    $$\lim_{z \to \pi}\ (z-\pi)\frac{1}{\sin z}.$$ Rewriting the limit in terms of ##z-\pi## is simply to make the evaluation easier. I just realized you could simply apply L'Hopital's rule to do that and avoid unnecessary complications. If you had a more complicated function, however, writing things in terms of ##z-\pi## is often less work than using L'Hopital's rule.

    But say you did it with using the trig identity anyway. You should end up with
    $$\lim_{z\to\pi} \frac{z-\pi}{-\sin(z-\pi)}.$$ You might already recognize that limit, but it not, use the substitution ##w=z-\pi## to turn it into one you should recognize.
     
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