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.
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)}
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}
ϕ
(
ϵ
)
≈
ϕ
(
0
)
{\displaystyle \phi (\epsilon )\approx \phi (0)}
(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 ]}
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]~.}}
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}
⟨
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}
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 .
Template:Lectures
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/828c799eb8842c7fa0a6e4fc3ad7bcc7a3fbcc2c)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7de076905153f9b3fd82011deb8e6b3ff027334e)
![{\displaystyle {\left\langle f',\phi \right\rangle =\left\langle {\text{pf}}\left[{\cfrac {H(x)}{x}}\right],\phi \right\rangle }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/684e69613ae7a9aa88c718aa9d3178648abd885c)
![{\displaystyle {\text{pf}}[\bullet ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e14af39ed57aa59b99a58e9b2de4750fc8fb6bb)
![{\displaystyle {f'(x)={\text{pf}}\left[{\cfrac {H(x)}{x}}\right]~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f05ab5b8df60ab4bc1c77b781b6b3ca9ec1a5bb)
![{\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]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e6306c15ef2f57b4cb7ef3dca6e56ad065d4f70)
![{\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]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/697295be24f28d3f795e8e441c926fb0e0f27037)
