= Meromorphic continuation of a Mellin transform from an asymptotic expansion
{title2=$\operatorname{Res}_{s=-\sigma_j}\mathcal Mf(s)=c_j$}
Suppose $f$ decays faster than every power at infinity and, near zero, $f(y)=\sum_{j=1}^r c_jy^{\sigma_j}+O(y^{\sigma_{r+1}})$ for every $r$, with $\sigma_j$ strictly increasing and tending to infinity. Subtracting finitely many terms in the integral over $(0,1)$ gives
$$
\mathcal Mf(s)=\int_1^\infty f(y)y^{s-1}\,dy+\sum_{j=1}^r\frac{c_j}{s+\sigma_j}+\int_0^1\left(f(y)-\sum_{j=1}^r c_jy^{\sigma_j}\right)y^{s-1}\,dy.
$$
The last integral is <holomorphic> on $\operatorname{Re}s>-\sigma_{r+1}$. These expressions continue the <Mellin transform> meromorphically to the whole plane, with simple poles precisely at $-\sigma_j$ for nonzero $c_j$.
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