Since the
th successive time derivative/integral of
Laplace-transforms to
, it follows that the
*fractional derivative* of order
, denoted by
, should correspond to the Laplace transform
, while the *fractional integral* of order
corresponds to
. We thus let
,
, denote both fractional integrals (
) and
derivatives (
), possibly including both an integer and
fractional part. However, since integer
corresponds to
a repeated ordinary derivative or integral, we will develop methods for
and implement any integer part in the
usual way. Expressing
as
so that
can be
defined as
, we obtain the corresponding Laplace and
Fourier transforms for fractional integration or differentiation as

Thus, a fractional integral or derivative of order corresponds to a phase shift by and a spectral ``tilt'' by . For , we obtain the frequency-response of a differentiator, and for , the integrator frequency-response is obtained, as required.

For integer
,
, we have, from the
*convolution theorem* for the *unilateral* Laplace transform
applied to *causal* functions
,^{4}

where ` ' denotes convolution, and denotes the Heaviside

This form was evidently developed originally as

where is any finite real number such that .

The generalization of
to
for
is
quite natural. There is also no problem extending
to
in Eq.(2), and the lower limit of
integration
can be extended (for all practical purposes)
as far as needed to the left to encompass the support of
. The
last piece is generalizing
to
, which is
provided by the *gamma
function*^{6}

which, for positive integers , becomes factorial,

for . We thus obtain the expression for fractional integrodifferentiation in the time domain as the convolution integral

This is known as the

where is an arbitrary fixed base point, and is any complex number with .

The topic of fractional differentiation and integration falls within
the well studied subject of *fractional
calculus*^{9}[1]. We will adopt the term ``differintegral''
from that literature.

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