What Lies Between a Function and Its Derivative? | Fractional Calculus

Can you take a derivative only partway? Is there any meaning to a "half-derivative"? Does such a concept even make sense? And if so, what do these fractional derivatives look like?


Previous video about Cauchy's Formula for Repeated Integration:
   • How to do two (or more) integrals with jus...  

A really nice video that derives the gamma function from scratch:
   • How to Take the Factorial of Any Number  


=Chapters=
0:00 - Interpolating between polynomials
1:16 - What should half derivatives mean?
3:56 - Deriving fractional integrals
8:22 - Playing with fractional integrals
9:12 - Deriving fractional derivatives
13:53 - Fractional derivatives in action
16:19 - Nonlocality
17:54 - Interpreting fractional derivatives
18:51 - Visualizing fractional integrals
22:10 - My thoughts on fractional calculus
23:10 - Derivative zoo


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MAIN SOURCES USED FOR THIS VIDEO

Podlubny, Igor. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, 1999

Podlubny, I.: "Geometric and physical interpretation of fractional integration and fractional differentiation." Fractional Calculus and Applied Analysis, vol. 5, no. 4, 2002, pp. 367--386.
(for the visualization trick for fractional integrals)

Edmundo Capelas de Oliveira, José António Tenreiro Machado, "A Review of Definitions for Fractional Derivatives and Integral", Mathematical Problems in Engineering, vol. 2014, Article ID 238459, 6 pages, 2014. https://doi.org/10.1155/2014/238459
(for the zoo of alternative fractional derivatives)


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Minor correction: The footnote at 7:34 should say the trig substitution produces another whole factor of pi (not a root pi) in the numerator which then cancels the two root(pi)'s that appear in the denominator from applying the half integral formula twice.


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CREDITS

This video uses the song "Rubix Cube" coming courtesy of Audionautix.com


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The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here:
https://github.com/morpho-matters/mor...


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This video is part of the 3Blue1Brown Summer of Math Exposition 2 (#SoME2). You can find out more about it here:
https://summerofmathexposition.substa...