18 I. Theory of Distributions
For any ε 0 we have
(5.15)
Rn
ϕ(x)dx
|x|2
(k ±
iε)2
=

0 |ω|=1
ϕ(rω)rn−1drdω
r2
(k ±
iε)2
.
Make the change of variables
r k = t, r = k + t.
Then
(5.16)
Rn
ϕ(x)dx
|x|2 (k ± iε)2
=

−k
ψε(t)dt,
t
where
(5.17) ψε(t) =
(k +
t)n−1
2k + t ±
|ω|=1
ϕ((k + t)ω)dω.
Analogously to Example 5.2 we get that the limit of (5.16) exists and
(5.18) lim
ε→+0
Rn
ϕ(x)dx
|x|2 (k ± iε)2
= p.v.

−k
ψ0(t)
t
dt ± iπψ0(0)
= p.v.
Rn
ϕ(x)dx
|x|2 k2
±
kn−2
2
|ω|=1
ϕ(kω)dω,
where by definition
(5.19) p.v.
Rn
ϕ(x)
|x|2

k2
dx = lim
δ→+0
||x|2−k2|δ
ϕ(x)dx
|x|2

k2
.
As in (5.6) we have:
p.v.
Rn
ϕ(x)dx
|x|2 k2
=
k
−k
ψ0(t) ψ0(0)
t
dt +

k
ψ0(t)
t
dt.
Since k 0, we have
1
|x|2 (k ± i0)2
=
1
|x|2 k2 i0
,
where
1
|x|2 k2 i0
= lim
ε→0
1
|x|2 k2
.
Let be the surface area element on the sphere
|x|2
=
k2.
Note that
=
kn−1dω.
Denote by
δ(|x|2−k2)
the distribution that acts on ϕ D by the formula:
(5.20)
|x|=k
1
2k
ϕ(x)dσ.
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