Applied linear operators and spectral methods/Differentiating distributions 2

Differentiation of distributions with severe discontinuities

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

Example 1

Let us first look at the distribution defined by the locally integrable function

$f(x)=H(x)~\log x$

where $H(x)$ is the Heaviside function.

Then

$\left\langle f,\phi \right\rangle =\int _{0}^{\infty }\phi (x)~\log(x)~{\text{d}}x$

By definition

$\left\langle f',\phi \right\rangle =-\left\langle f,\phi '\right\rangle =-\int _{0}^{\infty }\phi '(x)~\log(x)~{\text{d}}x$

Since the right hand side is a convergent integral, we can write

$\left\langle f',\phi \right\rangle =-\lim _{\epsilon \rightarrow 0}\int _{\epsilon }^{\infty }\phi '(x)~\log x~{\text{d}}x$

Integrating by parts,

$\left\langle f',\phi \right\rangle =-\lim _{\epsilon \rightarrow 0}\left[\int _{\epsilon }^{\infty }{\cfrac {\phi (x)}{x}}~{\text{d}}x+\phi (\epsilon )~\log(\epsilon )\right]$

Now, as $\epsilon \rightarrow 0$ we have $\phi (\epsilon )\approx \phi (0)$ and therefore

${\text{(1)}}\qquad \left\langle f',\phi \right\rangle \approx -\lim _{\epsilon \rightarrow 0}\left[\int _{\epsilon }^{\infty }{\cfrac {\phi (x)}{x}}~{\text{d}}x+\phi (0)~\log(\epsilon )\right]$

The right hand side of (1) gives a meaning to (i.e., regularizes) the divergent integral

$\int _{0}^{\infty }{\cfrac {\phi (x)}{x}}~{\text{d}}x~.$

We write

${\left\langle f',\phi \right\rangle =\left\langle {\text{pf}}\left[{\cfrac {H(x)}{x}}\right],\phi \right\rangle }$

where ${\text{pf}}[\bullet ]$ is the pseudofunction which is defined by the right hand side of (1).

In this sense, if

$f(x)=H(x)~\log x$

then

${f'(x)={\text{pf}}\left[{\cfrac {H(x)}{x}}\right]~.}$

Example 2

Next let the function to be differentiated be

$f(x)=\log |x|$

We can write this function as

$f(x)=H(x)~\log x+H(-x)~\log(-x)~.$

Then

${f'(x)={\text{pf}}\left[{\cfrac {H(x)}{x}}\right]+{\text{pf}}\left[{\cfrac {H(-x)}{x}}\right]={\text{pf}}\left[{\cfrac {1}{x}}\right]}$

where the pseudofunction $1/x$ is defined as the distribution

${\left\langle {\text{pf}}\left[{\cfrac {1}{x}}\right],\phi \right\rangle =\lim _{\epsilon \rightarrow 0}\left[\int _{\epsilon }^{\infty }{\cfrac {\phi (x)}{x}}~{\text{d}}x+\int _{-\infty }^{\epsilon }{\cfrac {\phi (x)}{x}}~{\text{d}}x\right]}$

The individual terms diverge at $\epsilon \rightarrow 0$ but the sum does not.

In this way we have assigned a value to the usually divergent integral

$\int _{\infty }^{\infty }{\cfrac {\phi (x)}{x}}~dx~.$

This value is more commonly known as the Cauchy Principal Value. Template:Lectures