An Introduction to Fractional Calculus

Does there exist such a thing as the half-derivative of a function? Can we find a way to interpolate between integer-order differentiation? The answers lie in the wonderful world of fractional calculus, which generalizes derivatives and integrals to any real (and even complex) number order. While it's a very pure field right now, some applications exist for a few physical problems and dynamic systems with non-local properties, as well as in quantum mechanics. Only time will tell how much it might expand in the distant future.

Timecodes:
0:00 - Intro and Nth Derivative Patterns
7:11 - Riemann-Liouville and Caputo Definitions
(correction: RL-Fractional integral won't converge for negative alpha**)
21:48 - Grunwald-Letnikov Definition
26:51 - Cauchy-Type Derivative and Definitions from Integral Transforms
32:46 - Product and Chain Rule for Fractional Derivatives
37:17 - A Few Applications and Closing

Sources and other tidbits:
Power function derivative interpolation Desmos animation: https://www.desmos.com/calculator/1da...
Cauchy's Formula for Repeated Integration: https://en.wikipedia.org/wiki/Cauchy_...
Fractional Calculus Overview: https://en.wikipedia.org/wiki/Fractio...
Differintegral operator: https://en.wikipedia.org/wiki/Differi...
Cauchy Fractional Derivative: https://www.researchgate.net/publicat...
Chain Rule for Fractional Derivatives: https://www.sciencedirect.com/science...
Fractional Calculus and the Tautochrone Problem: https://www.gcsu.edu/sites/files/page...
Modelling with Fractional Differential Equations: https://judekong.mathstats.yorku.ca/f...
Mittag-Leffler Function that shows up a lot in fractional differential equations: https://en.wikipedia.org/wiki/Mittag-...

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