Fractional integration

In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the q-differintegral of f, here denoted by

is the fractional derivative (if q > 0) or fractional integral (if q < 0). If q = 0, then the q-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several legitimate definitions of the differintegral.

The three most common forms are:

Recall the continuous Fourier transform, here denoted ${\displaystyle {\mathcal {F}}}$ :

Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:

So,

which generalizes to

Under the Laplace transform, here denoted by ${\displaystyle {\mathcal {L}}}$, differentiation transforms into a multiplication

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