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\centerline{{\it  Class 30} {\bf (Wednesday April 8)}}
 
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\item[SUMMARY] Using Laurent Series to Compute Residues
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\item[CURRENT READING] Brown \& Curchill pages 154--158
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\item[NEXT READING] Brown \& Curchill pages 210--219
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{\bf Laurent Series}\\
$$f(z) = \sum_{n=0}^{\infty} a_n (z-z_0)^n + \sum_{n=1}^{\infty} \frac{b_n}{(z-z_0)^n}, \qquad R_1 < |z-z_0| < R_2$$
This formula for a Laurent series is also sometimes written as
$$f(z) = \sum_{n=-\infty}^{\infty} c_n (z-z_0)^n \qquad \mbox{ where } c_n = \frac{1}{2 \pi i} \oint_C \frac{f(z)}{(z-z_0)^{n+1}} dz, \quad n = 0, \pm 1, \pm 2, \ldots$$
In practice, one usually computes a Laurent series by comparing the function you have to one of the ``famous functions'' whose Maclaurin series you already know.

\underline{\sc Groupwork}\\
{\bf 1.} Write down the Laurent series for $z^2 \sin(1/z^2)$ in the domain $0 < |z| < \infty$

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{\bf 2.} What is the value of Res$(z^2 \sin(1/z^2),0)$?
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{\bf 3.} Evaluate ${\displaystyle \oint_{|z|=1} z^2 \sin(1/z^2) dz}$ 
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{\bf 4.} Write down two different Laurent series for ${\displaystyle f(z) = \frac{1}{z(z^2+1)}}$ and specify the domain in which your series are valid.
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{\bf 5.} Evaluate ${\displaystyle \oint_{|z|=2} \frac{1}{z(z^2+1)} dz}$  using whichever method you prefer.
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