The asymptotic expansion of the swallowtail integral in the highly oscillatory region

摘要

We consider the swallowtail integral Psi (x, y, z) := integral(infinity)(-infinity) e(i(t5 +xt3) (+ yt2) (+zt)) dt for large values of vertical bar x vertical bar and bounded values of vertical bar y vertical bar and vertical bar z vertical bar. The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the simplified saddle point method introduced in (Lopez et al., 2009). The analysis is more straightforward with this method and it is possible to derive complete asymptotic expansions of Psi(x, y, z) for large vertical bar x vertical bar and fixed y and z. The asymptotic analysis requires the study of three different regions for arg x separated by three Stokes lines. The expansion is given in terms of inverse powers of x(1/3) and x(1/2) and the coefficients are elementary functions of y and z. The accuracy and the asymptotic character of the approximations is illustrated with some numerical experiments. (C) 2018 Elsevier Inc. All rights reserved.