In mathematical analysis, initialization of the differintegrals is a topic in fractional calculus, a branch of mathematics dealing with derivatives of non-integer order.
The composition law of the differintegral operator states that although:
wherein D<sup>−q</sup> is the left inverse of D<sup>q</sup>, the converse is not necessarily true:
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<nowiki>'</nowiki>(0) = C, ÃÂ<nowiki></nowiki>(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.
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 .