Applied linear operators and spectral methods/Differentiating distributions 2

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Differentiation of distributions with severe discontinuities[edit | edit source]

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

Example 1[edit | edit source]

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 2[edit | edit source]

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