If this is too broad, I apologise; let's keep it focused on the basics if necessary.
What's the motivation and the rigorous foundations behind fractional calculus?
It seems very weird & beautiful to me. Did it arise from some set of applications? If so (and even if not), here's a suitable question concerning its "physical meaning" and history.
The Wikipedia article makes it look quite clear-cut: stick $\Gamma$ into Cauchy's formula for repeated integration. But why can we do that? Why is it listed under "Heuristics"? I know the Gamma function generalises the factorial, but that's as much as I understand.
"Why ask?"
Because I like to see how different areas of Mathematics fit together. I like the way fractional calculus seems to take integration & differentiation and ask, "well, do we really need to do these things a natural number of times?" - and so on. So I'm just curious :)
