Initialized fractional calculus

In mathematical analysis, initialization of the differintegrals is a topic in fractional calculus.

Composition rule of Differintegrals
The composition law of the differintegral operator states that although:

$$\mathbb{D}^q\mathbb{D}^{-q} = \mathbb{I}$$

wherein D&minus;q is the left inverse of Dq, the converse is not necessarily true:


 * $$\mathbb{D}^{-q}\mathbb{D}^q \neq \mathbb{I}$$

Example
Consider elementary integer-order calculus. Below is an integration and differentiation using the example function $$3x^2+1$$:


 * $$\frac{d}{dx}\left[\int (3x^2+1)dx\right] = \frac{d}{dx}[x^3+x+C] = 3x^2+1\,,$$

Now, on exchanging the order of composition:


 * $$\int \left[\frac{d}{dx}(3x^2+1)\right] = \int 6x \,dx = 3x^2+C\,,$$

Where C is the constant of integration. Even if it was not obvious, the initialized condition &fnof; ' (0) = C, ƒ  (0) = D'', etc. could be used. If we neglected those initialization terms, the last equation would show the composition of integration, and differentiation (and vice versa) would not hold.

Description of initialization
Working with a properly initialized differ integral is the subject of initialized fractional calculus. If the differ integral is initialized properly, then the hoped-for composition law holds. The problem is that in differentiation, information is lost, as with C in the first equation.

However, in fractional calculus, given that the operator has been fractionalized and is thus continuous, an entire complementary function is needed. This is called complementary function $$\Psi$$.


 * $$\mathbb{D}^q_t f(t) = \frac{1}{\Gamma(n-q)}\frac{d^n}{dt^n}\int_0^t (t-\tau)^{n-q-1}f(\tau)\,d\tau + \Psi(x)$$