On Laplace transforms near the origin

Abstract:

Let $f(t)$ be locally integrable on $[0,\infty )$ and let $L\{ f\} (s)$ denote the Laplace transform of $f(t)$. In this note, we prove that if $f(t) \sim {t^{ - \beta }}\Sigma _{n = 0}^\infty {a_n}{(\log t)^{ - n}}$ as $t \to \infty$, where $0 \leqslant \operatorname {Re} \beta < 1$, then $L\{ f\} (s) \sim {s^{\beta - 1}}\Sigma _{n = 0}^\infty {c_n}{(\log 1/s)^{ - n}}$ as $s \to 0$ in $|\arg s| \leqslant \pi /2 - \Delta$, the ${c_n}$ being constants.