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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