  We study the family of Fourier-Laplace transforms $$ F_{\alpha,\beta}(z)=
\operatorname*{F.p.} \int_{0}^{\infty} t^{\beta}\exp(\mathrm{i}
t^{\alpha}-\mathrm{i} z t)\:\mathrm{d} t, \quad \operatorname*{Im} z<0, $$ for
$\alpha>1$ and $\beta\in\mathbb{C}$, where Hadamard finite part is used to
regularize the integral when $\operatorname*{Re} \beta\leq -1$. We prove that
each $F_{\alpha,\beta}$ has analytic continuation to the whole complex plane
and determine its asymptotics along any line through the origin. We also apply
our ideas to show that some of these functions provide concrete extremal
examples for the Wiener-Ikehara theorem and a quantified version of the
Ingham-Karamata theorem, supplying new simple and constructive proofs of
optimality results for these complex Tauberian theorems.
