# Can Laurent-Puiseux series be computed for annular regions?

## Main Question or Discussion Point

For example, consider:

$$f(z)=\sqrt{z(z-1)(z-2)}$$

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

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

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 $1<|z|<2$?

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

Not sure whether it will work here, but variable substitutions like $z \longmapsto \dfrac{1}{z}$ or $z \longmapsto 1 \pm z$ often help.