# Initialized fractional calculus

Part of a series of articles about :
Calculus :
$\int _{a}^{b}f'(t)\,dt=f(b)-f(a)$ :
Fundamental theorem
Limits
Continuity
Rolle's theorem
Mean value theorem
Inverse function theorem :
Differential
Definitions
Derivative (generalizations)
Differential
infinitesimal
of a function
total
Concepts
Differentiation notation
Second derivative
Implicit differentiation
Logarithmic differentiation
Related rates
Taylor's theorem
Rules and identities
Sum
Product
Chain
Power
Quotient
L'Hôpital's rule
Inverse
General Leibniz
Faà di Bruno's formula
Reynolds :
Integral
Lists of integrals
Integral transform
Leibniz integral rule
Definitions
Antiderivative
Integral (improper)
Riemann integral
Lebesgue integration
Contour integration
Integral of inverse functions
Integration by
Parts
Discs
Cylindrical shells
Substitution (trigonometric, tangent half-angle, Euler)
Euler's formula
Partial fractions (Heaviside's method)
Changing order
Reduction formulae
Differentiating under the integral sign
Risch algorithm :
Series
Geometric (arithmetico-geometric)
Harmonic
Alternating
Power
Binomial
Taylor
Convergence tests
Summand limit (term test)
Ratio
Root
Integral
Direct comparison
Limit comparison
Alternating series
Cauchy condensation
Dirichlet
Abel :
Vector
Gradient
Divergence
Curl
Laplacian
Directional derivative
Identities
Theorems
Gradient
Green's
Stokes'
Divergence
generalized Stokes
Helmholtz decomposition :
Multivariable
Formalisms
Matrix
Tensor
Exterior
Geometric
Definitions
Partial derivative
Multiple integral
Line integral
Surface integral
Volume integral
Jacobian
Hessian :
Advanced
Calculus on Euclidean space
Generalized functions
Limit of distributions :
Specialized
Fractional
Malliavin
Stochastic
Variations :
Miscellanea
Precalculus
History
Glossary
List of topics
Integration Bee
Mathematical analysis
Nonstandard analysis :
vte :
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:
$\mathbb {D} ^{q}\mathbb {D} ^{-q}=\mathbb {I}$
wherein D$^{−q}$ is the left inverse of D$^q$, 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 ƒ'(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 $\Psi$.
    $\mathbb {D} _{t}^{q}f(t)={\frac {1}{\Gamma (n-q)}}{\frac {d^{n}}{dt^{n}}}\int _{0}^{t}(t-\tau )^{n-q-1}f(\tau )\,d\tau +\Psi (x)$