Fractional derivative on circle

Question

I know the definition of a fractional derivative in x-axis in Fourier space:

$F(\partial^{\alpha}_x f(x) = -|k|^{\alpha} f(k)$

But I don't have any idea how to define fractional derivative on the boundary of a circle? Could anyone help me?

Answer 1

I'm not sure if this what you are after, but I am interpreting it as a desire to find the fractional derivative of a circle. I can accomplish this with ease in the complex plane. The details are explained in a previous post. There is a more extensive write-up here (see the linked PDF file).

The Cauchy pulse is defined as

$$\psi=\frac{1}{1-i\tau},\quad \tau\in(-\infty,\infty)$$

This function is amenable to the fractional calculus, as explained in the link and reference. If you plot $\psi$ in the complex plane you'll see that it is a circle. The equations in the link show how to get any real or complex fractional derivative (with an example).

Mathematically, the first derivative of $\psi$ is the cardioid. The fractional derivatives between, say

$$\frac{d^n\psi}{d\tau^n},\quad n\in[0,1]$$

exhibit a gradual change from the circle to the cardioid.

But the possibilities are endless.