Questions on the differentiation/integration of functions to fractional order. Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation and integration operators.

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 $D$ and integration operator $J$.

For example,

$$ \sqrt{D}=D^{\frac{1}{2}} $$

is an analog of the functional square root for the differentiation operator, i.e., an expression for some linear operator that when applied twice to any function will have the same effect as differentiation. More generally, one can look at the question of defining a linear functional $$ D^{a} $$ for every real-number $a$ in such a way that, when a takes an integer value $n ∈ ℤ$, it coincides with the usual $n$-fold differentiation $D$ if $n > 0,$ and with the $-n$–th power of J when $n < 0$.

It can be used in conjunction with the tag functional-analysis, partial-differential-equations, integral-operators.