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 ± iε

dx1 =

∞

−∞

x1 ∓ iε

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 ± iε

=

|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

=

eλ ln x1

for x1 0,

0 for x1 0,

x−

λ

=

0 for x1 0,

|x1|λ

=

eλ ln |x1|

for x1 0.

(5.11)

If Re λ −1, x+

λ

and x−

λ

define regular functionals. For Re λ ≤ −1, x±

λ

are

not integrable in a neighborhood of x1 = 0.

We define distributions regularizing x±

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.