Asymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integrals

We consider the highly oscillatory integral F ( w ) : = ∫ − ∞ e i t K + 2 θ p g d $F(w):=\int _{-\infty }^\infty e^{iw(t^{K+2}+e^{i\theta }t^p)}g(t)dt$ for large positive values of w, π < ≤ $-\pi <\theta \le \pi$ , and integers with 1 $1\le p\le K$ $g(t)$ an entire function. The standard saddle point method is complicated we use here a simplified version this introduced by López et al. derive asymptotic approximation when → $w\rightarrow +\infty$ general in terms elementary functions, determine Stokes lines. For ≠ $p\ne 1$ behavior may be classified four different regions according to even/odd character couple parameters p; special case $p=1$ requires separate analysis. As important application, family canonical catastrophe integrals Ψ x … $\Psi _K(x_1,x_2,\ldots ,x_K)$ one its variables, say $x_p$ bounded remaining ones. This written form $F(w)$ appropriate function . Then, | $\vert x_p\vert$ approximations are accompanied several numerical experiments. formulas presented fill up gap NIST Handbook Mathematical Functions Olver