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

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

## Answers and Replies

fresh_42
Mentor
Not sure whether it will work here, but variable substitutions like ##z \longmapsto \dfrac{1}{z}## or ##z \longmapsto 1 \pm z## often help.