FRACTIONAL ORDER OPERATORS

Causal integration is represented in the time domain by convolution with a step function. In the frequency domain this amounts to multiplication by $1/(-i\omega)$.Integrating twice amounts to convolution by a ramp function, $t\, {\rm step}(t)$, which in the Fourier domain is multiplication by $1/(-i\omega)^2$.Integrating a third time is convolution with $t^2\, {\rm step}(t)$ which in the Fourier domain is multiplication by $1/(-i\omega)^3$.In general

 

$$t^{n-1}\ {\rm step}(t) \quad =\quad{\rm FT}\ \left( { 1 \over (-i\omega)^n} \right)$$ (1)

Proof of the validity of equation (1) for integer values of n is by repeated indefinite integration which also indicates the need of an n! scaling factor. Proof of the validity of equation (1) for fractional values of n would take us far afield mathematically. (Fractional values of n, however, are exactly what we need to interpret Huygen's secondary wave sources in 2-D.) The factorial function of n in the scaling factor becomes a gamma function. The poles suggest that a more thorough mathematical study of convergence is warranted, but this is not the place for it.

A common application is when n=1/2 and (ignoring the scale factor) equation (1) becomes  

$${1\over \sqrt{t}} \ {\rm step}(t) \quad =\quad {\rm FT}\ \left( { 1 \over \sqrt{-i\omega}} \right)$$ (2)

It is well known that  

$${d \ \over dt} \quad =\quad {\rm FT}\ \left( -i\omega \right)$$ (3)

A product in the frequency domain corresponds to a convolution in the time domain. A time derivative is like convolution with a doublet $(1,-1)/\Delta t$.Thus, from equation (2) and equation (3) we obtain  

$${d \ \over dt} \ {1\over \sqrt{t}} \ {\rm step}(t) \quad =\quad {\rm FT}\ \left( \sqrt{-i\omega} \,\right)$$ (4)