Applied linear operators and spectral methods/Differentiating distributions 2

Differentiation of distributions with severe discontinuitiesEdit

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

Example 1Edit

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

 

where   is the Heaviside function.

Then

 

By definition

 

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

 

Integrating by parts,

 

Now, as   we have   and therefore

 

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

 

We write

 

where   is the pseudofunction which is defined by the right hand side of (1).

In this sense, if

 

then

 

Example 2Edit

Next let the function to be differentiated be

 

We can write this function as

 

Then

 

where the pseudofunction   is defined as the distribution

 

The individual terms diverge at   but the sum does not.

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

 

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