16 I. Theory of Distributions
We find the relation between (x1 ±
i0)−1
and p.v.
1
x1
. We have:
(5.8)

−∞
ϕ(x1)
x1 ±
dx1 =

−∞
x1
x1 2 + ε2
ϕ(x1)dx1 =
|x1| 1
x1(ϕ(x1) ϕ(0))
x1 2 + ε2
dx1
+
|x1| 1
x1ϕ(0)
x1
2
+ ε2
dx1 +
|x1| 1
x1
x1
2
+ ε2
ϕ(x1)dx1 i

−∞
εϕ(x1)dx1
x1
2
+ ε2
.
In the right hand side of (5.8) the second integral is equal to zero because
x1ϕ(0)
x1+ε22
is an odd function, the first and the third integrals converge as ε 0
by the Lebesgue convergence theorem, and the last integral in (5.8) is a
delta-like sequence (cf. Example 4.4, part b)).
Since

−∞
εϕ(x1)
x1 2 + ε2
dx1 = π

−∞
1
ε
β
x1
ε
ϕ(x1)dx1 πϕ(0),
where β(x1) =
1
π(x2+1)
1
, we get
lim
ε→+0

−∞
ϕ(x1)dx1
x1 ±
=
|x1| 1
ϕ(x1) ϕ(0)
x1
dx1 (5.9)
+
|x1| 1
ϕ(x1)
x1
dx1 iπϕ(0).
Comparing (5.9) and (5.6) we have
(5.10)
1
x1 ± i0
= p.v.
1
x1
iπδ.
Note that all three distributions
1
x1+i0
,
1
x1−i0
and p.v.
1
x1
are different regu-
larization of the nonintegrable function
1
x
.
Example 5.3 (Distributions x+
λ
and x−).
λ
For an arbitrary λ C, denote
x+
λ
=
x1
λ
=
ln x1
for x1 0,
0 for x1 0,
x−
λ
=
0 for x1 0,
|x1|λ
=
ln |x1|
for x1 0.
(5.11)
If Re λ −1, x+
λ
and x−
λ
define regular functionals. For Re λ −1,
λ
are
not integrable in a neighborhood of x1 = 0.
We define distributions regularizing
λ
for Re λ −1 by using the
method of analytic continuation in λ. The same method was originally used
to get the analytic continuation of the Gamma-function:
Γ(λ) =

0
x1−1e−x1 λ
dx1.
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