The Fourier Transform of the quartic Gaussian exp(-Ax4): Hypergeometric functions, power series, steepest descent asymptotics and hyperasymptotics and extensions to exp(-Ax2n)

摘要

The Fourier Transform of a quartic Gaussian, Φ(k)≡∫-∞∞exp(ikx)exp(-x4)dx, is important in the theory of radial basis functions with ϕ(r)=exp(-r4). We show that the transform is the exact sum of two generalized hypergeometric functions 0F2. This implies that Φ(k) is an entire function and also gives an analytical form for the power series coefficients of arbitrary degree. Using the method of steepest descent for integrals, we derive the first five terms in an asymptotic series in powers of k-4/3. Through a simple hyperasymptotic analysis, we show that the magnitude of the error of the optimally-truncated series [“superasymptotic” error] is Oexp-0.818k4/3. The lowest order approximation, Φ(k)∼27/6π/3k-1/3exp(-(21/33/16)k4/3)cos((33/221/3/16)k4/3-π/6) shows that the transform Φ(k) is an exponentially-decaying oscillation.We show that the steepest descent method [2,27] (Bender and Orszag, 1978; Miller, 2006) [26,33] (Lauwerier, 1966; Ursell, 1965) can be applied to the transform of exp(-Ax2n) for general integer n and give the first two terms.Another theme is that modern computer technology has simplified the steepest descent method. The steepest descent paths can be plotted by merely displaying the contours of the imaginary part of the phase function Ψ(t) where zΨ(t) is the logarithm of the integrand, and z is the large parameter (here proportional to k4/3). The task of inverting w(t) to obtain a series in powers of w for the metric factor dt/dw after the usual change of integration variable can now be done to high order by a few lines of an algebraic manipulation language as illustrated by a short table with the complete Maple code.