Meromorphic continuation of a Mellin transform from an asymptotic expansion (${\mathrm{Res}}_{s=-{\sigma }_j}\mathcal{M}f(s)=c_j$)

Suppose $f$ decays faster than every power at infinity and, near zero, $f(y)={\sum }_{j=1}^r{c}_j{y}^{{\sigma }_j}+O(y^{{\sigma }_{r+1}})$ for every $r$, with ${\sigma }_j$ strictly increasing and tending to infinity. Subtracting finitely many terms in the integral over $(0,1)$ givesThe last integral is holomorphic on $\mathrm{Re}s>-{\sigma }_{r+1}$. These expressions continue the Mellin transform meromorphically to the whole plane, with simple poles precisely at $-{\sigma }_j$ for nonzero $c_j$.

Strict increase alone does not suffice in the preceding theorem. Let ${\sigma }_j=1-1/j$, $c_j=2^{-j}$ and $f(y)=\eta (y){\sum }_{j\ge 1}{2}^{-j}{y}^{1-1/j}$, where $\eta$ is continuous, equals one on $(0,1]$ and vanishes on $[2,\infty )$. After division by the next power, every finite remainder extends continuously to zero. Nevertheless the Mellin transform has genuine poles at $-{\sigma }_j$ accumulating at $-1$, so it cannot be a meromorphic function on the whole plane.