# Specifying annular region while obtaining Laurent series

I'm trying to obtain the Laurent Series of $$f(z)=\frac{z}{(z+1)(z-2)}$$ about $$z=-1$$ in the annular region $$|z+1|>3$$. I've been trying to use the following code

Series[z/((z + 1) (z - 2)), {z, -1, 10}, Assumptions -> (Abs[z + 1] > 3)]


but it always returns the Laurent Series defined in the region $$0<|z+1|<3$$. Am I doing something incorrectly?

• So I'm not sure if this is quite right (my complex analysis here is a little rusty), but I think you can do Normal@Series[ z/((z + 1) (z - 2)) /. z -> z - 1 // Evaluate, {z, \[Infinity], 5}] /. z -> z + 1. This shifts the pole from -1 to 0, expands about infinity rather than 0, and then shifts z back to z-1. – march May 1 at 21:16

I think this can be done by expanding around infinity. This gives a power series in (for example z^-1). To get a power series in (z-1)^-1 requires a change of variable.

expr = z/((z + 1) (z - 2));

ser = Series[expr /. z -> w - 1, {w, \[Infinity], 5}]

(* SeriesData[w, DirectedInfinity[1], {1, 2, 6, 18, 54}, 1, 6, 1] *)

Normal[ser] /. w -> z + 1

(* 54/(1 + z)^5 + 18/(1 + z)^4 + 6/(1 + z)^3 + 2/(1 + z)^2 + 1/(1 + z) *)


I hope this is correct, my memory of Laurent series has faded a little.

• The code certainly supplies the same answer as I get when I do it by hand. Do you mind explaining in a bit more detail how it works, and how this technique would be used in general? – David May 1 at 21:32

A variation of mikado's answer is the following:

Series[z/((z + 1) (z - 2)) /. z->-1 + Defer[z+1], {Defer[z+1], Infinity, 10}] //TeXForm


$$\frac{1}{z+1}+\frac{2}{(z+1)^2}+\frac{6}{(z+1)^3}+\frac{18}{(z+1)^4}+\frac{54}{(z+1)^5}+\frac{162}{(z+1)^6}+\frac{486}{(z+1)^7}+\frac{1458}{(z+1)^8}+\frac{4374}{(z+1)^9}+\frac {13122}{(z+1)^{10}}+O\left(\left(\frac{1}{z+1}\right)^{11}\right)$$