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Laurent expansion for a complex function with 3 singularites

  1. Nov 28, 2013 #1
    1. The problem statement, all variables and given/known data
    Hey guys,

    So I need a bit of help with this question:

    Find three Laurent expansions around the origin, valid in three regions you should specify, for the function
    [itex]f(z)=\frac{30}{(1+z)(z-2)(3+z)}[/itex]


    2. Relevant equations
    None that I know of...just binomial expansion


    3. The attempt at a solution

    Okay so what I did was first specify the regions. Not sure if they are right though, although I think they are:

    Region 1: [itex]-1<|z|<2[/itex]
    Region 2: [itex]-3<|z|<-1[/itex]
    Region 3: [itex]|z|>2[/itex]

    Then I split f(z) into partial fractions:
    [itex]f(z)=\frac{2}{z-2}+\frac{3}{3+z}-\frac{5}{1+z}[/itex]

    Then I expanded for the region |z|>2, using the first term of the partial fractions (ignoring the other ones...right?) and I got

    [itex]f(z)_{z>2}=\Sigma_{n=1}^{\infty}(\frac{2}{z})^{n}[/itex]

    So now the problem is... first of all I dont know if that's right. Even if it is, I have no idea how to expand for the other regions...for example, say I wanted to do region 2...I dont even know where to start, do I first expand for -3<|z|, then |z|<-1 and add them...or what?

    Really need some help here guys! the fate of the universe hinges on this unfortunate question sheet!
     
  2. jcsd
  3. Nov 29, 2013 #2
    Let's start with the most obvious one: the regular Taylor series. Can you calculate it, and for what values of z does it converge?
     
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