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Finding the residue of a pole of order 2 (complex analysis)

  1. Jan 21, 2012 #1
    The problem

    Find Res(f,z1)

    With: [itex]f(z)=\frac{z}{(z^2+2aiz-1)^2}[/itex]

    The attempt at a solution

    The singularities are at [itex]A=i(-a+\sqrt{a^2-1})[/itex] and at [itex]B=i(-a-\sqrt{a^2-1})[/itex]

    With the normal equation (take limit z->A of [itex]\frac{d}{dz}((z-A)^2 f(z))[/itex] for finding the residue of a pole of order 2, my attempt fails.

    I know the way the correct answer is constructed, but I do not understand it.

    The solution is namely: take [itex]\frac{d}{dz}[/itex] of [itex]\frac{z}{(z-B)^2}[/itex]
    This is: [itex]\frac{-(z+B)}{(z-B)^3}[/itex]
    Then plug in z=A, which gives: [itex]\frac{-(A+B)}{(A-B)^3}[/itex]

    Now plugging in the values of A and B give the correct answer.

    Any help is appreciated.
     
    Last edited: Jan 21, 2012
  2. jcsd
  3. Jan 21, 2012 #2
    Never mind, I figured it out.
     
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