Fractional calculus

A natural question is whether the derivative and integral
can be defined for non-integer powers.
EG, is there a square root of the derivative?

It can be done, but it's -higher- analysis:
- higher derivatives are not infintesimal, or even local
  (the value at a point depends on the function globally)
- you likely want to use other transforms
  and the Gamma function

There are many ways to state this:
- Grunwald-Letnikov differintegral
  (yields derivative easily)
  uses the binomial version of higher derivatives;
  for natural numbers, this terminates,
  but for others it doesn't

- Riemann-Liouville differintegral
  Generalize Cauchy's formula for interated integral
  using the Gamma function

- Weyl differintegral
  (as R-L, but for periodic functions)

- Fourier/Laplace transform
  Just conjugate multiplication by $x^\alpha$;
  of course these transforms are not defined for everything.

Note that for sinusoids, fractional derivatives are exactly
translation (rotation on the circle).
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Amusing exercises:
ask students to conjecture/figure out what fractional derivatives
-should- be for:
- $\sin x$
- $x^n$
- $e^{ax}$
