Delta Functions

A Most Amiable Mapping

Firstly, I use the symbol delta because I couldn’t think of anything better to call it. Secondly, look up hyper4 operators and Conway exponentiation notation if you want to get into some of the proper maths around this. Also, I came across a lot of this stuff a few months ago in a facsimile of a really old calculus book (Back when there were people who differentiated for a living).

Define the function as
follows:

Here are some values of it:

Here are the main identities:

Proof of 1

This will be done by induction. Assume

holds for n. Then

,

but the exponent on the right can be rewritten
as

Which is precisely the form that we wanted.

Now, assume that the following holds for n+1:

Let us show that this implies that it holds for n also.

This proves the reverse case.

Setting n=0, we can see that it is true for any one value of n, and therefore by the first induction, for all n>0, and by the second, that it is true for all n<0.

This completes the proof of identity 1.

Proof of 2

This will also be done by 2-way induction.

Assume that

Holds for a particular n. Then

Which completes the positive induction. To carry out the negative induction, assume that

Holds for all n+1. Then

Which does the job. Letting n=0, we can see that

Which completes the proof.

These 2 rules can be simplified or made more complex to deal with various circumstances. For example:

or

i.e.

Anyway, those are all fairly trivial identities. Here’s a somehwat less trivial differential identity:

Proof

Assume the following true for a particular n

Then

Which shows the positive induction step.

Now, for the negative induction, assume the following to be true:

Now,

And finally, substitute n=1 into the original formula – first the hard(-ish) way

and now the easy way (and pray that they are equal)

And they are both equal. And so all is right with the world. Which proves it.

The formula simplifies into considerably more conceptually amusing versions, as

Or, more interestingly: