5. Regularizations of nonintegrable functions 15
5.1. Regularization in
R1.
Example 5.1 (Principal value integrals. Distribution p.v.
1
x1
). If g(x1) is
absolutely integrable on δ x1 +∞ for any δ 0, we define the principal
value integral, p.v.
∞
−∞
g(x1)dx1, as the limit of
−δ
−∞
g(x1)dx1+
∞
δ
g(x1)dx1
as δ → 0, assuming that this limit exists:
(5.4) p.v.
∞
−∞
g(x1)dx1 = lim
δ→0
x1δ
g(x1)dx1.
We show that for any ϕ ∈ D the principal value integral
p.v.
∞
−∞
1
x1
ϕ(x1)dx1
exists and defines a distribution that we denote by p.v.
1
x1
.
Indeed,
(5.5)
x1δ
ϕ(x1)dx1
x1
=
δx1 1
ϕ(x1) − ϕ(0)
x1
dx1
+
δx1 1
ϕ(0)
x1
dx1 +
x1 1
ϕ(x1)dx1.
x1
In the right hand side of (5.5) the second integral is equal to zero because
ϕ(0)
x1
is an odd function, and the first integral converges as δ → 0 since

ϕ(x1)−ϕ(0)
x1
 ≤ C.
Therefore the principal value integral exists and
(5.6) p.v.
∞
−∞
ϕ(x1)
x1
dx =
x1 1
ϕ(x1) − ϕ(0)
x1
dx +
x1 1
ϕ(x1)
x1
dx1.
It is clear that (5.5) defines a linear continuous functional on D. Note
that if ϕ(0) = 0 then the integrals
∞
0
ϕ(x1)
x1
dx1 and
0
−∞
ϕ(x1)
x1
dx1
diverge, but the principal value integral
p.v.
∞
−∞
ϕ(x1)
x1
dx1
exists.
Example 5.2 (Distributions
1
x1+i0
and
1
x1−i0
). These distributions were
already defined in Example 4.3, and they are derivatives in the sense of
distributions of regular functionals ln(x1 + i0) and ln(x1 − i0):
(5.7) (x1 ±
i0)−1
=
d
dx1
ln(x1 ± i0).