Asymptotic expansion from poles of a Mellin transform

Summary

The poles of the Mellin transform of a function $f$ control the behaviour of $f(x)$ as $x \to 0^+$ and as $x \to \infty$. A pole at $s = \rho$ left of the fundamental strip gives a term $x^{-\rho}$ in the asymptotic expansion of $f$ near $0$. A multiple pole adds powers of $\log x$ to this term. Poles to the right of the strip give the expansion of $f$ near $\infty$ in the same way. The method turns asymptotic questions about sums and integrals into the study of a meromorphic function. It is a basic tool in analytic number theory and in the analysis of algorithms.