14.3 Echoes 421
a similar procedure for Fourier transforms gives
F(g(x)h(x)) =
∫∞
−∞
g(x)h(x)eikxdx
=
∫∞
−∞
[
1
2 π
∫∞
−∞
̃g(k′)e−ik
′x
dk′
]
h(x)eikxdx
=
1
2 π
∫∞
−∞
dk′g ̃(k′)
∫∞
−∞
h(x)ei(k−k
′)x
dx
=
1
2 π
∫∞
−∞
dk′g ̃(k′) ̃h(k−k′). (14.125)
Thus, the Fourier-Laplace transform of Eq.(14.114) gives
(p+ikv)f ̃ 2 (p,k) +ik
e
m
∂f 0
∂v
̃φ 2 (p,k)
= −
e
m
∂
∂v
[∫
∞
−∞
dk′
2 π
∫b+i∞
b−i∞
dp′
2 π
k′ ̃φ 1 (p′,k′)f ̃ 1 (p−p′,k−k′)
]
.(14.126)
Because the convolution integrals are notationally unwieldy, to clarifythe notation we
define the new dummy variables
̄k′ = k−k′
̄p′ = p−p′ (14.127)
so that Eq.(14.126) becomes
(p+ikv)f ̃ 2 (p,k) +ik
e
m
∂f 0
∂v
φ ̃ 2 (p,k)
= −
e
m
∂
∂v
[∫
∞
−∞
dk′
2 π
∫b+i∞
b−i∞
dp′
2 π
k′φ ̃ 1 (p′,k′)f ̃ 1 ( ̄p′,k ̄′)
]
. (14.128)
The factors in the convolution integral can be expressed in terms of the original driving
potential using Eqs.(14.120) and (14.122) to obtain
(p+ikv)f ̃ 2 (p,k) +ik
e
m
∂f 0
∂v
φ ̃ 2 (p,k)=∂
∂v
χ(p,k,v) (14.129)
where the non-linear convolution term is
χ(p,k,v) =
(e
m
) 2 ∫∞
−∞
dk′
2 π
∫b+i∞
b−i∞
dp′
2 π
{
k′
φ ̃ext(p′,k′)
D(p′,k′)
×
i ̄k′
p ̄′+i ̄k′v
φ ̃ext( ̄p′, ̄k′)
D( ̄p′,k ̄′)
∂f 0
∂v