12 I. Theory of Distributions

Note that (x1 +

i0)λ

defines a regular functional if Re λ −1. By the

Lebesgue convergence theorem,

R1

(x1 +

iε)λϕ(x1)dx1

→

R1

(x1 +

i0)λϕ(x1)dx1

if ε → 0, Re λ −1, ∀ϕ ∈ D.

We denote by (x1 +

iε)λ

and (x1 +

i0)λ

the regular functionals corre-

sponding to integrable functions (x1 +

iε)λ

and (x1 +

i0)λ.

For any integer

m 0 and noninteger λ such that Re λ −1, we have

(x1 +

iε)λ−m

=

1

λ(λ − 1) · · · (λ − m + 1)

dm

dx1

m

(x1 +

iε)λ.

By Theorem 4.1, the distributions (x1 +

iε)λ−m

converge as ε → 0 to

the distribution

1

λ(λ−1)···(λ−m+1)

dm

dx1

m

(x1 +

i0)λ.

We shall denote the limiting distribution by (x1 +

i0)λ−m.

Therefore

(4.5) (x1 +

i0)λ−m

=

1

λ(λ − 1) · · · (λ − m + 1)

dm

dx1 m

(x1 +

i0)λ.

If λ = −k is an integer, k 0, then, starting with the identity

dk

dx1

k

ln(x1 + iε) =

(−1)(k−1)(k

− 1)!

(x1 + iε)k

, ε 0,

and using Theorem 4.1 and Example 4.1, we get that

(4.6) lim

ε→+0

(x1 +

iε)−k

=

(−1)k−1

(k − 1)!

dk

dx1

k

ln(x1 + i0) in D .

Therefore, for any λ ∈ C, there exists a limit of regular functionals

(x1 +

iε)λ

as ε → +0, and we denote this limiting distribution by (x1 +

i0)λ:

(4.7) lim

ε→+0

(x1 +

iε)λ

= (x1 +

i0)λ

in D , ∀λ ∈ C.

Analogously one defines (x1 −

i0)λ

as limε→+0(x1 −

iε)λ

in D , ∀λ ∈ C,

where (x1 −

iε)λ

= |x1 −

iε|λeiλ arg(x1−iε)

and −π arg(x1 − iε) 0. Thus

(x1 − i0)λ = |x1|λe−iλπθ(−x1).

4.1. Delta-like sequences.

Example 4.4.

a) Let β(x) be nonnegative and

Rn

β(x)dx = 1. Let

(4.8) βε(x) =

1

εn

β

x

ε

.

Then

(4.9) βε(ϕ) → δ(ϕ) = ϕ(0) as ε → 0, ∀ϕ ∈ D.