Fractional Calculus

Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.

and the integration operator J. (Usually J is used instead of I to avoid confusion with other I-like glyphs and identities.)

In this context the term powers refers to iterative application or composition, in the same sense that f 2(x) = f(f(x)).
For example, one may ask the question of meaningfully interpreting

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

for real-number values of a in such a way that when a takes an integer value n, the usual power of n-fold differentiation is recovered for n > 0, and the −nth power of J when n < 0.

The motivation behind this extension to the differential operator is that the semigroup of powers Da will form a continuous semigroup with parameter a, inside which the original discrete semigroup of Dn for integer n can be recovered as a subgroup. Continuous semigroups are prevalent in mathematics, and have an interesting theory. Notice here that fraction is then a misnomer for the exponent a, since it need not be rational; the use of the term fractional calculus is merely conventional.

Fractional differential equations are a generalization of differential equations through the application of fractional calculus.