You can take the derivative of a function once, twice, a hundred times. But what about half a time? Not half of the derivative, but rather the operation itself, applied to a degree of one half. It sounds like a joke. It isn't.
TIMESTAMPS:
0:00 The question — what would half a derivative even mean?
0:30 Derivatives of power functions and the pattern in the formula
1:15 The factorial problem — what is (½)!?
2:00 The Gamma function — definition and why it extends the factorial
2:55 Bohr-Mollerup — the Gamma function is the only option
3:40 Replacing factorials with Gamma to get a fractional derivative formula
4:20 Computing the half-derivative of x — a specific, graphable result
5:05 The composition check — two half-derivatives equal one full derivative
5:45 Riemann-Liouville — extending fractional calculus beyond power functions
6:35 Anomalous diffusion — where fractional derivatives govern real physics
7:20 Viscoelasticity — materials that live between integer derivative orders
8:05 The derivative as a dial, and the Gamma function as the mechanism
Music:
All music by Vincent Rubinetti
Piece 1: Reflections
Piece 2: Trinkets
Piece 3: Resonance
#FractionalCalculus #GammaFunction #Calculus #RiemannLiouville #AnomalousDiffusion #Mathematics #MathExplained
