The asymptotic behaviour of Pearcey’s integral for complex variables

Abstract
A deeper understanding of the rich structure of the canonical form of the oscillatory integral describing the cusp diffraction catastrophe, generally known as Pearcey’s integral P '( X , Y ), can be obtained by considering its analytic continuation to arbitrary complex variables X and Y . A new integral representation for P '( X , Y ) is given in the form of a contour integral involving a Weber parabolic cylinder function whose order is the variable of integration. It is shown how the asymptotics of P '( X , Y ) may be obtained from this representation for complex X and Y when either | X | or | Y | → ∞, without reference to the usual stationary points of the integrand. For the case | X | → ∞, Y finite the full asymptotic expansion of P '( X , Y ) is derived and its asymptotic character is found to be either exponentially large or algebraic in certain sectors of the X - plane. The case | Y | → ∞ , X finite is complicated by the presence of exponentially small subdominant terms in certain sectors of the Y - plane, and only the first terms in the expansion are given. The asymptotic behaviour of P '( X , Y ) on the caustic Y 2 + (2/3 X ) 3 = 0 is also obtained from the new representation and is shown to agree with recent results of D. Kaminski. The various properties of the Weber parabolic cylinder function required in this paper are collected together in the Appendix.

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