  Asymptotic expansions are obtained for contour integrals of the form \[
\int_a^b \exp \left( - zp(t) + z^{\nu /\mu } r(t) \right)q(t)dt, \] in which
$z$ is a large real or complex parameter, $p(t)$, $q(t)$ and $r(t)$ are
analytic functions of $t$, and the positive constants $\mu$ and $\nu$ are
related to the local behaviour of the functions $p(t)$ and $r(t)$ near the
endpoint $a$. Our main theorem includes as special cases several important
asymptotic methods for integrals such as those of Laplace, Watson, Erd\'elyi
and Olver. Asymptotic expansions similar to ours were derived earlier by Dingle
using formal, non-rigorous methods. The results of the paper also serve to
place Dingle's investigations on a rigorous mathematical foundation. The new
results have potential applications in the asymptotic theory of special
functions in transition regions, and we illustrate this by two examples.
