Initialized fractional calculus

In mathematical analysis, initialization of the differintegrals is a topic in fractional calculus, a branch of mathematics dealing with derivatives of non-integer order.

Composition rule of Differintegrals

The composition law of the differintegral operator states that although:
wherein D^−q is the left inverse of D^q, the converse is not necessarily true:

Example

Consider elementary integer-order calculus. Below is an integration and differentiation using the example function :
Now, on exchanging the order of composition:
Where C is the constant of integration. Even if it was not obvious, the initialized condition ƒ' = C, ƒ = D'', etc. could be used. If we neglected those initialization terms, the last equation would show the composition of integration, and differentiation 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.