In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the q-differintegral of f, here denoted by
D q f is the fractional derivative (if q > 0) or fractional integral (if q < 0). If q = 0, then the q-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several legitimate definitions of the differintegral.
Standard definitions
The three most common forms are:
The Riemann–Liouville differintegralThis is the simplest and easiest to use, and consequently it is the most often used. It is a generalization of the
Cauchy formula for repeated integration to arbitrary order.
a D t q f ( t ) = d q f ( t ) d ( t − a ) q = 1 Γ ( n − q ) d n d t n ∫ a t ( t − τ ) n − q − 1 f ( τ ) d τ The Grunwald–Letnikov differintegralThe Grunwald–Letnikov differintegral is a direct generalization of the definition of a derivative. It is more difficult to use than the Riemann–Liouville differintegral, but can sometimes be used to solve problems that the Riemann–Liouville cannot.
a D t q f ( t ) = d q f ( t ) d ( t − a ) q = lim N → ∞ [ t − a N ] − q ∑ j = 0 N − 1 ( − 1 ) j ( q j ) f ( t − j [ t − a N ] ) The Weyl differintegralThis is formally similar to the Riemann–Liouville differintegral, but applies to
periodic functions, with integral zero over a period.
Recall the continuous Fourier transform, here denoted F :
F ( ω ) = F { f ( t ) } = 1 2 π ∫ − ∞ ∞ f ( t ) e − i ω t d t Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:
F [ d f ( t ) d t ] = i ω F [ f ( t ) ] So,
d n f ( t ) d t n = F − 1 { ( i ω ) n F [ f ( t ) ] } which generalizes to
D q f ( t ) = F − 1 { ( i ω ) q F [ f ( t ) ] } . Under the Laplace transform, here denoted by L , differentiation transforms into a multiplication
L [ d f ( t ) d t ] = s L [ f ( t ) ] . Generalizing to arbitrary order and solving for Dqf(t), one obtains
D q f ( t ) = L − 1 { s q L [ f ( t ) ] } . Linearity rules
D q ( f + g ) = D q ( f ) + D q ( g ) D q ( a f ) = a D q ( f ) Zero rule
D 0 f = f Product rule
D t q ( f g ) = ∑ j = 0 ∞ ( q j ) D t j ( f ) D t q − j ( g ) In general, composition (or semigroup) rule is not satisfied:
D a D b f ≠ D a + b f D q ( t n ) = Γ ( n + 1 ) Γ ( n + 1 − q ) t n − q D q ( sin ( t ) ) = sin ( t + q π 2 ) D q ( e a t ) = a q e a t 