**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 −*n*th power of *J* when *n* < 0.

The motivation behind this extension to the differential operator is that the semigroup of powers *D**a* will form a *continuous* semigroup with parameter *a*, inside which the original *discrete* semigroup of *D**n* 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.

Read more about Fractional Calculus: Nature of The Fractional Derivative, Heuristics, Fractional Derivative of A Basic Power Function, Laplace Transform, Fractional Derivatives, Functional Calculus

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