Differentiation of distributions with severe discontinuities
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Now we will consider the case of differentiation of some locally continuous
integrable functions whose discontinuities are more severe than simple jumps.
Example 1
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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
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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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