Part of a series of articles about
Calculus
${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
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
v t e

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader. (May 2024) (Learn how and when to remove this message) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Initialized fractional calculus" – news · newspapers · books · scholar · JSTOR (May 2024) (Learn how and when to remove this message) This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (May 2024) (Learn how and when to remove this message) (Learn how and when to remove this message)

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:

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

wherein D^−q is the left inverse of D^q, the converse is not necessarily true:

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

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

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

Now, on exchanging the order of composition:

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

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.

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 ${\displaystyle \Psi }$.

${\displaystyle \mathbb {D} _{t}^{q}f(t)={\frac {1}{\Gamma (n-q)}}{\frac {d^{n}}{dt^{n}}}\int _{0}^{t}(t-\tau )^{n-q-1}f(\tau )\,d\tau +\Psi (x)}$

• Initial conditions
• Dynamical systems

• Lorenzo, Carl F.; Hartley, Tom T. (2000), Initialized Fractional Calculus (PDF), NASA (technical report).