Asymptotic Expansion of the Pearcey Integral Near the Caustic

Abstract

An asymptotic expansion of the function \[ P(x,y) = \int_R {\exp \left\{ {i\left( {{{t^4 } / 4} + {{xt^2 } / 2} + yt} \right)} \right\}dt} \] is constructed which remains uniformly valid as $x \to - \infty $ for $(x,y)$ near the caustic $4x^3 + 27y^2 = 0$ in the real $xy$-plane. The result remains valid for a range of complex x and y when P is extended to ${\bf C} \times {\bf C}$ by analytic continuation.