Use partial fractions and then geometric series.
Hey, I'm having a lot of trouble with this Laurent expansion because it's a bit more complicated than the ones I'm used to. Sorry it's not in LaTeX.
f(z) = z / (z - 2i)(z + 1)
valid for (i) 0 < |z + 1| < sqrt(5)
(ii) |z| > 2
Thanks for any help!
For (i) You can use the substitution that with some steps transforms the function...
(1)
... into ...
(2)
Now if is...
(3)
... so that if You turn back to the z variable and consider that is...
(4)
... from (3) with some other steps you obtain finally the Laurent expansion...
(5)
For (ii) You set and proceed is similar way... the computation is left to You...
Kind regards