Asymptotic Analysis and Singular Perturbation Theory

The standard normalization for the Airy function corresponds to c = 1/(2π), and
thus
Ai(x) = 1
∞
e
2π −∞
i(kx+k3 /3) dk. (3.13)
This oscillatory integral is not absolutely convergent, but it can be interpreted
as the inverse Fourier transform of a tempered distribution. The inverse transform
is a C∞ function that extends to an entire function of a complex variable, as can
be seen by shifting the contour of integration upwards to obtain the absolutely
convergent integral representation
Ai(x) = 1
∞+iη
e
2π −∞+iη
i(kx+k3 /3) dk.
Just as the Fresnel integral with a quadratic phase, provides an approximation
near a nondegenerate stationary phase point, the Airy integral with a cubic phase
provides an approximation near a degenerate stationary phase point in which the
third derivative of the phase in nonzero. This occurs when two nondegenerate
stationary phase points coalesce.
Let us consider the integral
I(x, ε) =
∞
−∞
f(x, t)e iϕ(x,t)/ε dt.
Suppose that we have nondegenerate stationary phase points at
t = t±(x)
for x < x0, which are equal when x = x0 so that t±(x0) = t0. We assume that
ϕt (x0, t0) = 0, ϕtt (x0, t0) = 0, ϕttt (x0, t0) = 0.
Then Chester, Friedman, and Ursell (1957) showed that in a neighborhood of (x0, t0)
there is a local change of variables t = τ(x, s) and functions ψ(x), ρ(x) such that
ϕ(x, t) = ψ(x) + ρ(x)s + 1
3 s3 .
Here, we have τ(x0, 0) = t0 and ρ(x0) = 0. The stationary phase points correspond
to s = ± −ρ(x), where ρ(x) < 0 for x < x0.
Since the asymptotic behavior of the integral as ε → 0 is dominated by the
contribution from the neighborhood of the stationary phase point, we expect that
I(x, ε) ∼
∞
−∞
1
i[ψ(x)+ρ(x)s+
f (x, τ(x, s)) τs(x, s)e 3 s3 ]/ε
ds
∼ f (x0, t0) τs(x0, 0)e iψ(x)/ε
39
∞
−∞
1
i[ρ(x)s+
e 3 s3 ]/ε
ds