Integrals of the type

${\int}_{{\mathcal{C}}_{n}}g\left(z\right)exp(-kf\left(z\right))dz$ are considered, where

$k$ is a large parameter. It is assumed that

$f$ has several first order saddle points (simple zeros of

${f}^{\text{'}}\left(z\right)$).

${\mathcal{C}}_{n}$ is an infinite contour through the saddle point

${z}_{n}$. With the aid of the principle of resurgence, inspired by the works of Dingle and Écalle, the divergence of the asymptotic expansions is interpreted in terms of the influence of other saddle points. Hyperasymptotic methods are used in order to obtain an improved asymptotic expansion. The term ‘hyperasymptotics’ is introduced in an earlier paper by

*M. V. Berry* and

*C. J. Howls* [ibid. 430, No. 1880, 653-668 (1990;

Zbl 0745.34052)] used for functions defined by differential equations.