Editing Initialized fractional calculus (section)

== Composition rule of Differintegrals ==

The [[Differintegral#Basic formal properties|composition law]] of the [[differintegral]] operator states that although: 

<math>\mathbb{D}^q\mathbb{D}^{-q} = \mathbb{I}</math>

wherein ''D''<sup>&minus;''q''</sup> is the left [[Inverse function|inverse]] of ''D<sup>q</sup>'', the converse is not necessarily true:

:<math>\mathbb{D}^{-q}\mathbb{D}^q \neq \mathbb{I}</math>

===Example===

Consider elementary integer-order [[calculus]]. Below is an integration and differentiation using the example function <math>3x^2+1</math>:

:<math>\frac{d}{dx}\left[\int (3x^2+1)dx\right] = \frac{d}{dx}[x^3+x+C] = 3x^2+1\,,</math>

Now, on exchanging the order of composition:

:<math>\int \left[\frac{d}{dx}(3x^2+1)\right] = \int 6x \,dx = 3x^2+C\,,</math>

Where ''C is'' the [[constant of integration]]. Even if it was not obvious, the initialized condition ''&fnof;''<nowiki>'</nowiki>(0)&nbsp;=&nbsp;''C'', ''ƒ''<nowiki>''</nowiki>(0)&nbsp;=&nbsp;''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.