As far as I've seen, it turns out that some definitions of the half-derivative do actually have some use in the real world.
"What's a half derivative?" you ask. It's something that if you apply it twice against a function, you obtain the derivative of the original function.
Likewise, the half integral can be defined directly by some method or as the half derivative of a full integral.
As for learning about it, the basics are relatively simple if you're familiar with even the most basic calculus.
Be prepared to meet the Gamma function, if you are not familiar with that. If you've done any calculus, you might have noticed that the factorial turns up pretty often. The Gamma function is the standard interpolation of the factorial for non-integers, and there are closed form results for half integers.
Half integer "factorials" turn out to be the key to accessing half integer derivatives, etc.
YouTube has many videos on the topic.
As for getting into using it in the real world, I have no idea. I just enjoy watching the videos occasionally.