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) · · · 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) · · · 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.
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