An extension of Laplace's method

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.