# 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

${\displaystyle f(x)=H(x)~\log x}$

where ${\displaystyle H(x)}$ is the Heaviside function.

Then

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

By definition

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

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

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

Integrating by parts,

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

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

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

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

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

We write

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

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

In this sense, if

${\displaystyle f(x)=H(x)~\log x}$

then

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

### Example 2

Next let the function to be differentiated be

${\displaystyle f(x)=\log |x|}$

We can write this function as

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

Then

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

where the pseudofunction ${\displaystyle 1/x}$ is defined as the distribution

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

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

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

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

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