Smoothing of the Higher‐Order Stokes Phenomenon

ABSTRACT For over a century, the Stokes phenomenon had been perceived as a discontinuous change in the asymptotic representation of a function. In 1989, Berry demonstrated it is possible to smooth this discontinuity in broad classes of problems with the prefactor for the exponentially small contribution switching on/off taking a universal error function form. Following pioneering work of Berk, Nevins, and Roberts and the Japanese school of formally exact asymptotics, the concept of the higher‐order Stokes phenomenon was introduced, whereby the ability for the exponentially small terms to cause a Stokes phenomenon may change, depending on the values of parameters in the problem, corresponding to the associated Borel‐plane singularities transitioning between Riemann sheets. Until now, the higher‐order Stokes phenomenon has also been treated as a discontinuous event. In this paper, we show how the higher‐order Stokes phenomenon is also smooth and occurs universally with a prefactor that takes the form of a new special function, based on a Gaussian convolution of an error function. We provide a rigorous derivation of the result, with examples spanning the gamma function, a second‐order nonlinear ODE, and the telegraph equation, giving rise to a ghost‐like smooth contribution present in the vicinity of a Stokes line, but which rapidly tends to zero on either side. We also include a rigorous derivation of the effect of the smoothed higher‐order Stokes phenomenon on the individual terms in the asymptotic series, where the additional contributions appear prefactored by an error function.