Fractional calculus

Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator $D$

and of the integration operator $J$

and developing a calculus for such operators generalizing the classical one.

In this context, the term powers refers to iterative application of a linear operator to a function, in some analogy to function composition acting on a variable, i.e. $f \circ2 (x) = f \circ f (x) = f (f (x) )$ .

For example, one may ask the question of meaningfully interpreting

as an analogue of the functional square root for the differentiation operator, i.e., an expression for some linear operator that when applied twice to any function will have the same effect as differentiation. More generally, one can look at the question of defining a linear functional

for every real-number values of the parameter $a$ in such a way that, when $a$ takes an integer value $n \in ℤ$ , it coincides with the usual $n$ -fold differentiation $D$ if $n > 0$ , and with the $-n$ –th power of $J$ when $n < 0$ .

One of the motivations behind the introduction and study of such kind of extensions of the differentiation operator $D$ is that the sets of operator powers ${D a | a \in ℝ$ } defined in this way are continuous semigroups with parameter $a$ , of which the original discrete semigroup of ${D n | n \in ℤ$ } for integer $n$ is a denumerable subgroup: since continuous semigroups have a well developed mathematical theory, it is interesting to apply it to other branches of mathematics.

...
Wikipedia