Later the Delta Function was named after their founder. Dirac himself called it an improper function.

This time, the poem is not from repurposed snippets of his prose. These are just my own words to describe a proof:

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**In the limit**

the Lorentzian becomes

the improper function.

In the limit of tiny epsilons

it becomes infinitely thin.

Thus under the integral

it singles out

another function’s value at zero.

Yet the area covered

by this delicate needle

is finite and unity.

This is the property

we need to assert.

To integrate over the Lorentzian bell,

we consider the argument

a complex number.

Note the poles on the imaginary axis!

The spike peters out

at infinite distance,

regardless of angle.

We can safely add a tour

through the upper imaginary half plane.

It would add nothing

to the integral.

The path has been closed

outlining a half moon of sorts.

It lassos the upper pole!

Shrink the infinite half moon

into a small loop around the pole.

Trace out the circular path.

Parameterize: Crank the phase angle!

Exponential functions cancel,

and the pole is defused.

A tiny radius and a tiny epsilon:

Both will become zero in the limit.

Let the radius evaporate first!

Then epsilons cancel out,

as well as the imaginary units.

We are left with a number of one half.

Integrating over the circle just gives

two times the Ludolphian number.

Numbers cancel.

We obtain unity.

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