The Airy integral.
Take the differential equation (
)
Ansatz: Consider a
Laplace-type integral
for some function 
and contour 
independent of 
.
Then
Plug into the
ODE:
We want this to vanish for all , so we try to make the integrand a total derivative in 
. We’d like
for some 
. Compute:
Thus,
Matching terms gives
so
Solving,
Hence,
Now we check:
Then
But
so
The ODE is satisfied provided the term
vanishes —or, preferably, decays—at both
ends of .
Set
so
For large , the dominant term is 
. Its behavior at infinity is determined by 
.
We have decay when
Let 
. Then 
. From 
we get
Define three sectors:
Sector 1 (around 
, sector’s bisector is 
):
Sector 2
(around 
, sector’s bisector is 
):
Sector 3
(around 
, sector’s bisector is 
):
Each sector has width 
.
To find the saddle points we
differentiate:
Thus, there
are two saddle points. Writing 
and solving 
we obtain
For 
, this gives
Hence,
We seek for constant phase curves, i.e., 
. On those curves we must have decay
as goes to infinity.
Two curves pass through each saddle point,
but only one enters sectors 1, 2 and 3 at the endpoints.
Let 
where 
. Do a Taylor expansion:
Now compute
Locally
Write 
. So:
Along a steepest-descent path, the
condition must hold. Hence, for the path passing through
(so 
), the steepest descent curve satisfies
From this
The condition implies
Hence
Since the directions repeat every 
, the distinct cases correspond to 
. The resulting directions are
These are
approximate: take 
for 
and 
for 
.
The decay contours for are shown in the
following plot as 
and 
. Therefore,
solves the differential equation for 
or 
.
As the following plot suggests,
reflecting 
yields a third contour (yellow) running from
sector 1 below the x‑axis to sector 3 below its bisector.
Thus, the
integrals
for 
all solve the differential equation
Since this is second order, only two
solutions are linearly independent, so the integrals must be related.
There is no singularity of the integrand
within 
, so Cauchy’s theorem yields
Thus,
We define
This is the Airy integral which
was studied by Airy in relation to diffraction near caustic surface.
Diffraction near a caustic surface occurs
when light rays “pile up’’ along an envelope of rays, producing intense,
rapidly varying brightness patterns that cannot be described by simple
geometric optics alone.
It is exactly in this regime that the Airy integral appears: it is the
universal mathematical model for diffraction near a fold caustic. A fold
caustic is a stable, bright line pattern formed by light rays converging
and reflecting or refracting off a curved, irregular surface. It represents the
boundary or "envelope" where light rays become tangent, creating a
high-intensity, "folded" edge in the, for example, light patterns
seen on the bottom of a swimming pool.
In optics and mathematics, an envelope is
a curve or surface that is tangent to every member of a family of other curves.
If we put 
, i.e., we reduce the path to the imaginary axis,
It still converges.
Indeed, let
then
Take 
. Split
The first term is finite so it suffices
to study the tail.
Integrating by parts on 
,
Notice that
for large 
, 
, hence
Also,
so for 
Thus,
converges as an improper oscillatory
integral, and therefore so does its real part,
We have
established that the oscillatory integral defining the Airy function is indeed
convergent, providing a solid analytical foundation for its integral
representation. This result allows us to treat the Airy function with
confidence and opens the door to a deeper exploration of its properties.
In the next
episode—the final chapter of Season 2—we move beyond convergence and uncover the asymptotic behavior of
Airy functions, extracting precise formulas that reveal their behavior for
large arguments.
And if you’re
enjoying this journey through analysis, consider picking up The Riemann Hypothesis Revealed for a broader perspective on the fascinating world of
modern mathematics.
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