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# 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 .Integrating twice amounts to convolution by a ramp function, , which in the Fourier domain is multiplication by .Integrating a third time is convolution with which in the Fourier domain is multiplication by .In general

 (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
 (2)

It is well known that
 (3)
A product in the frequency domain corresponds to a convolution in the time domain. A time derivative is like convolution with a doublet .Thus, from equation (2) and equation (3) we obtain
 (4)

Next: HANKEL TAIL Up: Claerbout: Hankel tail Previous: INTRODUCTION
Stanford Exploration Project
11/17/1997