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 {\displaystyle D} D f ( x ) = d d x f ( x ) , {\displaystyle Df(x)={\frac {d}{dx}}f(x)\,,} and of the integration operator J {\displaystyle J} J f ( x ) = ∫ 0 x f ( s ) d s , {\displaystyle Jf(x)=\int _{0}^{x}f(s)\,ds\,,} 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 D {\displaystyle D} to a function f {\displaystyle f} , that is, repeatedly composing D {\displaystyle D} with itself, as in D n ( f ) = ( D ∘ D ∘ D ∘ ⋯ ∘ D ⏟ n ) ( f ) = D ( D ( D ( ⋯ D ⏟ n ( f ) ⋯ ) ) ) .