I'm just learning Laurent series. Could someone help me with this:
Find the Laurent series for the function z/[(z+1)(z+2)] in the following regions:
|z| < 1, 1 < |z| < 2, and |z| < 2.
Laurent expansions about z = 0, I suppose?
With partial fractions, you can rewrite:
z/((z+1)(z+2)) = 2/(z+2) - 1/(z+1)
You can now use the known Taylor series for |z| < 1. Watch out, you'll need a subtle manipulation for the other intervals, in order to make sure that the Taylor series still converge.