
1260 SOME ASYMPTO TIC RESULTS AND FORMULAS
3
◦
. For a function ϕ(x
0
, t) supported in a neighborhood of the point (x
0∗
, t
∗
), define the
function
[
e
Kφ](x, t, ~) =
¯
F
1/~
p→x
e
i
~
S(p,t)
ϕ(x
0
, t)
p
|J
1
(x
0
, t)|
x
0
=x
0
(p,t)
(x, t, ~), (20.2.1.18)
where
¯
F
1/~
p→x
is the Fourier transform with parameter ~ defined below in Section 20.2.3.
That is what the solution should look like near focal points. So far, we have only one
problem: for this function to satisfy the equation, the function φ(x
0
, t) should be indepen-
dent of t, but then the expression (20.2.1.18) is not defined, because the function x
0
(p, t)
is not defined everywhere on the support of ϕ. This problem will ...