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.



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




Example 2Edit

Next let the function to be differentiated be


We can write this function as




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