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### Fractional Derivatives and Integrals

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

 (1)

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

 (2)

where  ' denotes convolution, and denotes the Heaviside unit step function:

This form was evidently developed originally as Cauchy's repeated integral formula.5

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 function6

which, for positive integers , becomes factorial, i.e.,

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

This is known as the Riemann-Liouville differintegral,7more commonly stated closer to the following form:

where is an arbitrary fixed base point, and is any complex number with .8

The topic of fractional differentiation and integration falls within the well studied subject of fractional calculus9[1]. We will adopt the term differintegral'' from that literature.

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