 Solution

Fix a small $s>0$, for example $s<\sigma /2$, independently of $t$. Taking imaginary parts of the identity in part (a) givesThe parameter-shift remainder bound for a logarithmic Laplace integral makes the last term $O(t^{-\sigma -s})$, smaller than $t^{-\sigma }{L}^{-k}$ for every fixed $k$.

The endpoint function $h$ is analytic near zero. For any fixed $N$, its Taylor expansion isIntegrating this expansion against $e^{-Lr}$ useswhile the error is $O(L^{-N-2})$ and the extension from $s$ to infinity contributes an exponentially small tail. This is Watson's lemma, and givesEquivalently the coefficients have the stated exponential generating function $h(r)={\sum }_{n\ge 0}{\beta }_n{r}^n/n!$. In particular,Thus the apparent $t^{-\sigma }(\log t{)}^{-1}$ term vanishes: the first nonzero terms are $t^{-\sigma }[\Gamma (\sigma )(\log t{)}^{-2}+2{\Gamma }^{\prime }(\sigma )(\log t{)}^{-3}+\cdots ]$. The derivatives of the gamma function supply the successive logarithmic coefficients.