SMOOTHING OF THE STOKES PHENOMENON FOR HIGH-ORDER DIFFERENTIAL-EQUATIONS

We extend the class of functions for which the smooth transition of a Stokes multiplier across a Stokes line can be rigorously established to functions satisfying a certain differential equation of arbitrary order n. The equation chosen admits solutions of hypergeometric function type which, in the case n = 2, are related to the parabolic cylinder functions. In general, the solutions of this equation involve compound asymptotic expansions, valid in certain sectors of the complex z-plane, with more than one dominant and subdominant series. The functional form of the Stokes multipliers, expressed in terms of an appropriately scaled variable describing transition across a Stokes line, is found to obey the error function smoothing derived by Berry.