Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers of the differential operator

$D = \frac{d}{dx} \,$

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

In this context powers refer to iterative application or composition, in the same sense that f2(x) = f(f(x)).
For example, one may pose the question of interpreting meaningfully

$\sqrt{D} = D^{1/2} \,$

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

$D^s \,$

for real-number values of s in such a way that when s 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.

There are various reasons for looking at this question. One is that in this way the semigroup of powers Dn in the discrete variable n is seen inside a continuous semigroup (one hopes) with parameter s which is a real number. Continuous semigroups are prevalent in mathematics, and have an interesting theory. Notice here that fraction is then a misnomer for the exponent, since it need not be rational, but the termfractional calculus has become traditional.