### Fractional Calculus

Take the differential operator $$\frac{d}{dx}$$, which can be applied to functions (e.g. $$f(x)$$). In some instances, we want to take the derivative multiple times (e.g. $$\frac{d}{dx}(\frac{d}{dx}f(x))$$), which we abbreviate with exponents to save space (e.g. $$\frac{d^2}{dx^2}$$). This gives us the generalized operation $$\frac{d^n}{dx^n}, \forall n \in \mathbb{N}$$. If you think about it, this seems a bit like exponentiation - a repeated operation denoted as a superscript. We know that exponents can be negative, so we might wonder if we could create derivatives of a negative order (e.g. $$\frac{d^{-1}}{dx^{-1}}$$). In fact, negative derivatives are called anti-derivatives, and are the same as integrals (up to a constant). This extends our derivative operator to $$n \in \mathbb{Z}$$. We acknowledge the relationship between integration and differentiation by calling them both differintegrals, where derivatives perform positive differintegration and integration performs negative differintegration. However, this answer will hardly satisfy a curious reader. Why are we limited to integers? What other sets can $$n$$ belong to? $$\mathbb{Q}$$? $$\mathbb{R}$$? In fact, we can have differintegrals of any order within $$\mathbb{C}$$! This seems crazy, but so did complex exponentiation before its relation to trig functions was recognized: $$\cos x = \frac{e^{ix}+e^{-ix}}{2}$$. I’m still learning about fractional calculus myself, but so far it’s been fun and thought provoking!

This GIF (wikipedia) illustrates fractional derivatives $$\frac{d^n}{dx^n}, n \in [-1,1]$$ for $$f(x)=x$$.

Let’s Learn, Nemo! has a really great Youtube playlist on the subject, and so far I’ve found The Fractional Calculus (Oldham) to be a pretty good book. They provide most of theorems without included proofs, which forces you to gain insights by proving theorems as you read.