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Can Laurent-Puiseux series be computed for annular regions?

  1. Nov 13, 2011 #1
    For example, consider:

    [tex]f(z)=\sqrt{z(z-1)(z-2)}[/tex]

    It's easy to compute the Laurent-Puiseux series in the unit disc, up to the singular point at z=1:

    [tex]f(z)=\sqrt{2} \sqrt{z}-\frac{3 z^{3/2}}{2 \sqrt{2}}-\frac{z^{5/2}}{16 \sqrt{2}}-\frac{3 z^{7/2}}{64 \sqrt{2}}-\frac{37 z^{9/2}}{1024 \sqrt{2}}+\cdots,\quad |z|<1[/tex]

    That's done by creating a differential equation for the function with polynomial coefficients then solving it using power series where in this case, the indical equation has a root c=1/2 to generate the fractional powers.

    But can we generate a Laurent-Puiseux series for the function in the annular region [itex]1<|z|<2[/itex]?

    I haven't found any info on the net about this and was hoping someone here could shed some light on the matter.

    Thanks,
    Jack
     
  2. jcsd
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