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Can someone share how to find a Laurent series expansion of $$f(z)=\frac{1}{(z^2-1)(z^2-4)}$$ centered at zero on the annular disk $1<|z|<2$?

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You can use Apart to break the function to simpler rational expressions but I don't know if there is an automated command to give you the series for such cases. –  Spawn1701D Apr 19 '13 at 3:04
    
It is not clear from this question as it is posed, whether it is a math question or a Mathematica question. Please clarify. –  m_goldberg Apr 19 '13 at 4:51
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1 Answer

One slick way to compute the coefficients $c_k$ in the Laurent series

$$f(z)=\sum_{k\in\mathbb Z} c_k (z-a)^k$$

is to recognize that the problem of computing them is equivalent to the problem of computing Fourier coefficients, if you take the contour $\gamma$ in the definition for Laurent coefficients,

$$c_k=\frac1{2\pi i} \oint_\gamma \frac{f(z)\,\mathrm dz}{(z-a)^{k+1}}$$

to be a circle of radius $r$ within the annulus of interest. In your case we can take $r=3/2$, so the computation of the coefficients can be done like so:

With[{r = 3/2, n = 8},
     Table[FourierCoefficient[With[{z = r Exp[I t]}, 1/((z^2 - 1) (z^2 - 4))], t, k]/r^k,
          {k, -n, n, 2}]]
   {-1/3, -1/3, -1/3, -1/3, -1/12, -1/48, -1/192, -1/768, -1/3072}

(Exercise: why did I skip the computation of the odd-indexed coefficients?)

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Answer to exercise: You were recording a video –  belisarius Apr 19 '13 at 12:38
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@bel, did you not sign the non-disclosure agreement? –  J. M. Apr 19 '13 at 12:52
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