As in title I'm trying to write the Laurent series of for , but I'm stuck...
Ok, another last thing, the exercise was at the beginning: Laurent series of for ,
then I wrote partial fractions: and then I wrote the Laurent Series for every fraction:
At this point my question is: I can leave and as they are?
So the final result will be:
If it is allowed to consider a Taylor expansion a particular case of Laurent expansion where all the negative coefficients are 0, then the expansion of around ...
(1)
... is a Laurent expansion. The (1) converges for but with a procedure called 'analytic extension' the analyticity of can be 'extended' to the region . Of course a spontaneous question is: what's the practical utility of all that?...
Kind regards
The final answer is correct, but it looks a bit clumsy because of the two separate summations, which could easily be combined into one. In fact, it would have been better to write the partial fraction expression using only one term with denominator
That way, you get a constant term 1 for the Laurent series, plus a single series arising from the z-1 denominator.