The abscissa of absolute summability of Laplace integrals

Abstract:

With $A(u)$ of bounded variation over every finite interval of the nonnegative real axis, we write $C(w) = \smallint _0^w{e^{ - us}}dA(u)$ and (formally) \[ R(k’,w) = {(\Gamma (k’ + 1))^{ - 1}}\int _w^\infty {{{(u - w)}^{k’}}\;dA(u)\quad (k’ \geqq 0).} \] It is shown that if k is positive and fractional and if ${e^{ - ws’}}R(k,w)$ is summable $|C,0|$ for some $s’$ whose real part is negative, then $C(w)$ is summable $|C,k + \varepsilon |$ for each $\varepsilon > 0$, where s is such that its real part is greater than that of $s’$; if k is nonnegative and integral the result holds with $\varepsilon = 0$. Together with a ’converse’ result, this may be used to show that if the abscissa of $|C,k|$ summability of $\smallint _0^\infty {e^{ - us}}dA(u)$ is negative then it equals \[ \lim \sup \limits _{w \to \infty } {w^{ - 1}}\log \int _w^\infty {|dR(k,u)|} \] for all $k \geqq 0$ except one fractional value.