Differentiation of distributions with severe discontinuities
edit
Now we will consider the case of differentiation of some locally continuous
integrable functions whose discontinuities are more severe than simple jumps.
Example 1
edit
Let us first look at the distribution defined by the locally integrable
function
f
(
x
)
=
H
(
x
)
log
x
{\displaystyle f(x)=H(x)~\log x}
where
H
(
x
)
{\displaystyle H(x)}
is the Heaviside function.
Then
⟨
f
,
ϕ
⟩
=
∫
0
∞
ϕ
(
x
)
log
(
x
)
d
x
{\displaystyle \left\langle f,\phi \right\rangle =\int _{0}^{\infty }\phi (x)~\log(x)~{\text{d}}x}
By definition
⟨
f
′
,
ϕ
⟩
=
−
⟨
f
,
ϕ
′
⟩
=
−
∫
0
∞
ϕ
′
(
x
)
log
(
x
)
d
x
{\displaystyle \left\langle f',\phi \right\rangle =-\left\langle f,\phi '\right\rangle =-\int _{0}^{\infty }\phi '(x)~\log(x)~{\text{d}}x}
Since the right hand side is a convergent integral, we can write
⟨
f
′
,
ϕ
⟩
=
−
lim
ϵ
→
0
∫
ϵ
∞
ϕ
′
(
x
)
log
x
d
x
{\displaystyle \left\langle f',\phi \right\rangle =-\lim _{\epsilon \rightarrow 0}\int _{\epsilon }^{\infty }\phi '(x)~\log x~{\text{d}}x}
Integrating by parts,
⟨
f
′
,
ϕ
⟩
=
−
lim
ϵ
→
0
[
∫
ϵ
∞
ϕ
(
x
)
x
d
x
+
ϕ
(
ϵ
)
log
(
ϵ
)
]
{\displaystyle \left\langle f',\phi \right\rangle =-\lim _{\epsilon \rightarrow 0}\left[\int _{\epsilon }^{\infty }{\cfrac {\phi (x)}{x}}~{\text{d}}x+\phi (\epsilon )~\log(\epsilon )\right]}
Now, as
ϵ
→
0
{\displaystyle \epsilon \rightarrow 0}
we have
ϕ
(
ϵ
)
≈
ϕ
(
0
)
{\displaystyle \phi (\epsilon )\approx \phi (0)}
and
therefore
(1)
⟨
f
′
,
ϕ
⟩
≈
−
lim
ϵ
→
0
[
∫
ϵ
∞
ϕ
(
x
)
x
d
x
+
ϕ
(
0
)
log
(
ϵ
)
]
{\displaystyle {\text{(1)}}\qquad \left\langle f',\phi \right\rangle \approx -\lim _{\epsilon \rightarrow 0}\left[\int _{\epsilon }^{\infty }{\cfrac {\phi (x)}{x}}~{\text{d}}x+\phi (0)~\log(\epsilon )\right]}
The right hand side of (1) gives a meaning to (i.e., regularizes) the
divergent integral
∫
0
∞
ϕ
(
x
)
x
d
x
.
{\displaystyle \int _{0}^{\infty }{\cfrac {\phi (x)}{x}}~{\text{d}}x~.}
We write
⟨
f
′
,
ϕ
⟩
=
⟨
pf
[
H
(
x
)
x
]
,
ϕ
⟩
{\displaystyle {\left\langle f',\phi \right\rangle =\left\langle {\text{pf}}\left[{\cfrac {H(x)}{x}}\right],\phi \right\rangle }}
where
pf
[
∙
]
{\displaystyle {\text{pf}}[\bullet ]}
is the pseudofunction which is defined by
the right hand side of (1).
In this sense, if
f
(
x
)
=
H
(
x
)
log
x
{\displaystyle f(x)=H(x)~\log x}
then
f
′
(
x
)
=
pf
[
H
(
x
)
x
]
.
{\displaystyle {f'(x)={\text{pf}}\left[{\cfrac {H(x)}{x}}\right]~.}}
Example 2
edit
Next let the function to be differentiated be
f
(
x
)
=
log
|
x
|
{\displaystyle f(x)=\log |x|}
We can write this function as
f
(
x
)
=
H
(
x
)
log
x
+
H
(
−
x
)
log
(
−
x
)
.
{\displaystyle f(x)=H(x)~\log x+H(-x)~\log(-x)~.}
Then
f
′
(
x
)
=
pf
[
H
(
x
)
x
]
+
pf
[
H
(
−
x
)
x
]
=
pf
[
1
x
]
{\displaystyle {f'(x)={\text{pf}}\left[{\cfrac {H(x)}{x}}\right]+{\text{pf}}\left[{\cfrac {H(-x)}{x}}\right]={\text{pf}}\left[{\cfrac {1}{x}}\right]}}
where the pseudofunction
1
/
x
{\displaystyle 1/x}
is defined as the distribution
⟨
pf
[
1
x
]
,
ϕ
⟩
=
lim
ϵ
→
0
[
∫
ϵ
∞
ϕ
(
x
)
x
d
x
+
∫
−
∞
ϵ
ϕ
(
x
)
x
d
x
]
{\displaystyle {\left\langle {\text{pf}}\left[{\cfrac {1}{x}}\right],\phi \right\rangle =\lim _{\epsilon \rightarrow 0}\left[\int _{\epsilon }^{\infty }{\cfrac {\phi (x)}{x}}~{\text{d}}x+\int _{-\infty }^{\epsilon }{\cfrac {\phi (x)}{x}}~{\text{d}}x\right]}}
The individual terms diverge at
ϵ
→
0
{\displaystyle \epsilon \rightarrow 0}
but the sum does not.
In this way we have assigned a value to the usually divergent integral
∫
∞
∞
ϕ
(
x
)
x
d
x
.
{\displaystyle \int _{\infty }^{\infty }{\cfrac {\phi (x)}{x}}~dx~.}
This value is more commonly known as the Cauchy Principal Value .
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