The asymptotic expansion of certain canonical integrals

Integrals of the form $\text{I(λ) = ∝}{}_{\mathrm{R}}{}^2\text{?(}\text{x}\text{?}\text{)}\text{exp}\text{{iλφ(}\text{x}\text{?}\text{)} d}\text{x}\text{?}$, common in diffraction theory, are usually evaluated asymptotically for large λ using the stationary phase technique. At caustic points the classical technique is not applicable. A computation-oriented algorithm is developed for determining the asymptotic expansion of such integrals with $\text{φ(}\text{x}\text{)}$ a Thom umbilic, the canonical form appropriate at caustic points for most cases of physical interest.