4. Convergence of distributions 11

Example 4.1. If fn(x) → f(x) a.e. and |fn(x)| ≤ g(x) for all n, where g(x)

is a locally integrable function, then, by the Lebesgue convergence theorem,

fn(ϕ) =

Rn

fn(x)ϕ(x)dx →

Rn

f(x)ϕ(x)dx = f(ϕ), ∀ϕ ∈ D.

In particular, consider

(4.2) ln(x1 + iε) = ln |x1 + iε| + i arg(x1 + iε),

where ε 0 and 0 arg(x1 + iε) π for all x1 ∈

R1.

Denote

(4.3) ln(x1 + i0) = lim

ε→+0

ln(x1 + iε) = ln |x1| + iπθ(−x1),

where θ(−x1) = 0 for x1 0 and θ(−x1) = 1 for x1 0. Since | ln(x1+iε)| ≤

π + 1 + | ln |x1|| for |x1| ≤ 1, and | ln(x1 + iε)| ≤ π + ln ||x1| + 1| for |x1| ≥

1, 0 ε 1, we get by the Lebesgue convergence theorem that the regular

functionals, corresponding to ln(x1 + iε), converge as ε → 0 to the regular

functional, corresponding to ln(x1 + i0).

Example 4.2. Denote fn(x1) = n if 0 x1

1

n

, and fn(x1) = 0 if x1 does

not belong to [0,

1

n

]. Then

∞

−∞

fn(x1)dx1 = 1 and

∞

−∞

fn(x1)ϕ(x1)dx1 − ϕ(0) = n

1

n

0

(ϕ(x1) − ϕ(0))dx1

≤ max

0≤x1≤

1

n

|ϕ(x1) − ϕ(0)| → 0

as n → ∞. Therefore

fn(ϕ) → δ(ϕ)

as n → ∞.

Theorem 4.1. If fm → f in D , then for any k = (k1,...,kn),

∂kfm

∂xk

→

∂kf

∂xk

in D .

Proof: We have, by the definition of the derivative of a distribution:

∂kfm

∂xk

(ϕ) =

(−1)|k|fm(

∂kϕ

∂xk

) →

(−1)|k|f(

∂kϕ

∂xk

) =

∂kf

∂xk

(ϕ).

Example 4.3. Let λ be a complex number. Consider

(x1+iε)λ

=

eλ ln(x1+iε),

where ε 0 and ln(x1 + iε) is defined as in Example 4.1. Thus

(x1 +

iε)λ

= |x1 +

iε|λeiλ arg(x1+iε),

0 arg(x1 + iε) π.

Denote

(4.4) (x1 +

i0)λ

= lim

ε→+0

(x1 +

iε)λ

=

|x1|λeiλπθ(−x1).