given the following function
$$ f(y)= \int_{0}^{\infty}dx \frac{2}{(x+y)(x+y+1)} $$
i wish to expand $ f(y) $ into a Laurent series
is then valid if i use a numerical approxiamtion for the integral over 'x' so
$$ f(y)= \sum_{j}c_{j}\frac{2}{(x_{j}+y)(x_{j}+y+1)}$$
and then i exapnd each summand of the sum (Which depend only on 'y') to get the Laurent series for $ f(y) $
The improper integral $f(y)$ under consideration equals $ 2\ln(y+1)-2\ln(y) $ for each complex number $y$ which does not belong to $\{y:\Im y=0,\,\Re y <0\}$. Because $y=0$ is not an isolated singularity of $f(y)$, the Laurent expansion of $f(y)$ at $y=0$ does not exist.