Differintegral

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In mathematics, the combined differentiation/integration operator used in fractional calculus is called the differintegral, and it has a few different forms which are all equivalent, provided that they are initialized (used) properly.

It is noted:

and is most generally defined as:

Table of contents

1 Standard definitions
2 Definitions via transform
3 History
4 Web Resources
5 Book resourcees

Standard definitions

By far, the three most common forms are:

The Riemann-Liouville differintegral(RL)

This is the simplest and easiest to use, and consequently it is the most often used.

We first introduce the Riemann-Liouville fractional integral, which is a straight-forward generalization of the Cauchy formula for repeated integration:

This gives us integration to an arbitrary order. To get differentation to an arbitrary order, we simply integrate to arbitrary order n-q, and differentiate the result to integer order n. (We choose n and q so that n is the smallest positive integer greater than or equal to q (that is, the ceiling of q)):

Thus, we have differentiated n-(n-q)=q times. The RL differintegral is thus defined as(the constant is brought to the front):

definition

When we are taking the differintegral at the upper bound (t), it is usually written:

definition

And when we are assuming that the lower bound is zero, it is usually written:

definition

That is, we are taking the differintegral of f(t) with respect to t.

see for more info: Riemann-Liouville differintegral.

The Grunwald-Letnikov differintegral(GL). -This has an interesting form. It may provide some geometric insight into fractional calculus if we can develop a good intepretation of it. This also poses fewer restrictions on the function being differintegrated. It is a generalization of the infinite Riemann Sum which defines integer-order integration in calculus. It uses the Binomial coefficient (generalized by the gamma function to arbitrary domain), commonly used in the branch of mathematics called counting.:

definition

see for more info: Grunwald-Letnikov differintegral.

The Weyl differintegral. -This is very similar to the Riemann-Louiville differintegral. The most important difference is that its upper bound is infinity.

see for more info: Weyl differintegral.

Definitions via transform

Any function can be defined in a space isomorphic to a space which it has been shown to be defined in. We therefore define the differintegral via its behavior in certain transformed spaces corresponding to some common transformations.

The Fourier Transform. -Firstly, we define differintegration in Fourier space (using the continuous Fourier transform, here denoted F). In Fourier space, differentation transforms into a simple translation:

This easily generalizes to: