What the Heck is a Half Derivative?
When we think of a derivative, what exactly is the thing we think of?
Note: this article is a flyover bird’s eye view of some elementary fractional calculus. Most of the material is stuff I covered in my undergraduate thesis, which you can DM me for if you are interested.
This article assumes knowledge of integral and differential calculus, and some basic algebra skills.
The Naive Symbolic Approach
There are a few models for the derivative: the limit of the average slope (the old finite difference formula:
The derivative is also the thing that “undoes” the integral, a sort of inverse integral, as shown by the (cue boomy voice) Fundamental Theorem of Calculus (part 1 and 2):
So, then, say we wanted to have some kind of algebraic way to look at these. Say, we wanted to really nail down that idea of “inverse” derivative, “inverse” integral. We could start with something nice like this:
In my way of thinking, this is the sort of object the derivative and integral are in relationship. Let’s nail this down a little further, and say we can define multiple derivatives and their inverses:
Now, n being an integer is a bit restrictive when I write it like this, don’t you think? Yeah! But how would that work? The way it’s written still corresponds to something we understand:
We might also want something like the power rule for derivatives to hold, which can happen with the Gamma function (discussed next):
The exponents could easily be fractional on the proper domain, but the factorial takes a little bit of work.
The Slightly Smarter (But Still Slightly Wrong) Approaches
Now, if you have followed my other articles (such as the one on complex analysis), you might know that there is a special way to define the derivative through the integral. Don’t worry about the notation; the integral is an integral around a contour in the complex plane, which we are going to skim over. This is the Cauchy…