Taking Fractional Derivatives In the Frequency Domain

Going to make me do all the heavy lifting, are you?

Proof:

multiplying each amplitude by its frequency

multiplying each amplitude by its frequency^2

Since there shouldn't be anything quirky going on:

multiplying each amplitude by its frequency^(3/2)

So just raise each frequency to the order of the fractional derivative.

Incredibly easy!

A Stanford youtube math prof said that people think in the time domain but nature operates in the frequency domain.

You can really see it here.

Bret Cahill

This is the launchpoint for the book _Calculus in a New Key_ by D. L. Orth. It sure SOUNDS interesting, but the noninteger derivative doesn't simplify anything I've ever worked on.

It's good to check your work.

If you want to do it with the Excel Fourier transform tool:

1. get the new phase angle from the derivative order, nu
2. get the real and imaginary parts from the new phase angle and the absolute value of the transform and the sign of the original real and imaginary parts.
3. multiply the phase adjusted real and imaginary by the frequency^nu
4. complex

1. inverse transform.

Bret Cahill

Bonjour,

I do not know the purpose and use of 'your' fractional derivative. to sum up (d/dy)^r o exp(ay) = a^r*exp(ay) (d/dy)^r o y^n = Gamma(n+1)/Gamma(n-r+1)*y^(n-r) We may also directly build a function g(x,y) such as g(x+r,y) = d/dy)^r o g(x,y) Example g(x,y) = d/dy)^x o (exp(2y)+exp(3y) g(x,y) = 2^x*exp(2y)+3^x*exp(3y) ,

Alain

If a function cannot be represented analytically, a spreadsheet is a convenient way to quickly look at a lot of different fractional derivatives.

I'm still having some divide by zero issues. Excel can sometimes be tricked into working just by using very small numbers for zero.

Bret Cahill