  The value of a highly oscillatory integral is typically determined
asymptotically by the behaviour of the integrand near a small number of
critical points. These include the endpoints of the integration domain and the
so-called stationary points or saddle points -- roots of the derivative of the
phase of the integrand -- where the integrand is locally non-oscillatory.
Modern methods for highly oscillatory quadrature exhibit numerical issues when
two such saddle points coalesce. On the other hand, integrals with coalescing
saddle points are a classical topic in asymptotic analysis, where they give
rise to uniform asymptotic expansions in terms of the Airy function. In this
paper we construct Gaussian quadrature rules that remain uniformly accurate
when two saddle points coalesce. These rules are based on orthogonal
polynomials in the complex plane. We analyze these polynomials, prove their
existence for even degrees, and describe an accurate and efficient numerical
scheme for the evaluation of oscillatory integrals with coalescing saddle
points.
