i have been reading about Laurent series. Their formal definition is given as
$$ f(z) = \sum_{n=1}^\infty \frac {b_n} {(z-a)^n} + \sum_{n=0}^\infty a_n(z-a)^n $$
Where $ b_n = \frac { 1}{2i\pi}\int_\Gamma f(w)(w-a)^{n-1} dw $
and $ a_n =\frac {1}{2i\pi}\int_\Gamma \frac{f(w)}{(w-a)^{n+1}} dw $
Pretty much everywhere i read about this ALL examples just use manipulations of geometric series to find the Laurent expansion. However i was looking to see if anyone could educate me specifically on which contour is used, if i were to decide to use contour integration to find the $a_n$ and $b_n$ coefficients.
Lets say for example finding the series expansion of $ \frac {1}{1-z} $ valid for $ |z| > 1 $ which is $$ - \sum_{n=1}^\infty \frac{1}{z^n} $$ but using contour integration please.
I apologize if this is not worded properly. Thank you very much in advance for your help educating me and your time.