*From Wikipedia, the free encyclopedia*

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

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 *f*^{2}(x) = f(f(x)).

For example, one may pose the question of interpreting meaningfully

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 *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 −*n*th 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 *D*^{n} 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 term*fractional calculus* has become traditional.

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