Differential equations for the cuspoid canonical integrals

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Abstract
Differential equations satisfied by the cuspoid canonical integrals I n (a) are obtained for arbitrary values of n≥2, where n−1 is the codimension of the singularity and a=(a 1,a 2,...,a n−1). A set of linear coupled ordinary differential equations is derived for each step in the sequence I n (0,0,...,0,0) →I n (0,0,...,0,a n−1) →I n (0,0,...,a n−2,a n−1) →...→I n (0,a 2,...,a n−2,a n−1) →I n (a 1,a 2,...,a n−2,a n−1). The initial conditions for a given step are obtained from the solutions of the previous step. As examples of the formalism, the differential equations for n=2 (fold), n=3 (cusp), n=4 (swallowtail), and n=5 (butterfly) are given explicitly. In addition, iterative and algebraic methods are described for determining the parameters a that are required in the uniform asymptotic cuspoid approximation for oscillating integrals with many coalescing saddle points. The results in this paper unify and generalize previous researches on the properties of the cuspoid canonical integrals and their partial derivatives.