Differintegral

In fractional calculus, an area of mathematical analysis, 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.
Standard definitions.
The four most common forms are:
Definitions via transforms.
Recall the continuous Fourier transform, here denoted formula_8 :
Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:
So,
which generalizes to
Under the bilateral Laplace transform, here denoted by formula_13 and defined as formula_14, differentiation transforms into a multiplication
Generalizing to arbitrary order and solving for "D""q""f"("t"), one obtains
Basic formal properties.
"Linearity rules"
"Zero rule"
"Product rule"
In general, "composition (or semigroup) rule" is not satisfied: