# Fractional Definite Integral

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Definite Fractional Integrals

#### 2.1. On the One-Sided Integer Order Derivatives and Their Inverses

#### 2.2. Order 1 Integral

#### 2.3. Definite Fractional Integral

**Definition**

**1.**

#### 2.4. Which Fractional Derivative?

**criterion if it enjoys the properties:**

^{1}P^{1}P1- Linearity
^{1}P2- Identity
^{1}P3- Backward compatibility
^{1}P4- The index law holds for negative orders
^{1}P5- Generalised Leibniz rule

**keeps four conditions and**

^{2}P**is modified to:**

^{1}P4^{2}P4 The index law

#### 2.5. The Riemann-Liouville and Caputo Derivatives

- The RL-FD is an inverse operator of the left RL-FI$${}^{RL}\phantom{\rule{-2.84544pt}{0ex}}{D}_{a+}^{\alpha}{}^{RL}\phantom{\rule{-2.84544pt}{0ex}}{I}_{a+}^{\alpha}f(x)=f(x)$$
- The C-FD is also an inverse operator of the left RL-FI$${}^{C}\phantom{\rule{-2.84544pt}{0ex}}{D}_{a+}^{\alpha}{}^{RL}\phantom{\rule{-2.84544pt}{0ex}}{I}_{a+}^{\alpha}f(x)=f(x).$$

- If $f(x)\in {L}_{1}(a,b)$ and ${f}_{n-\alpha}(x)={I}_{a+}^{n-\alpha}f(x)\in A{C}^{n}[a,b]$, then$${}^{RL}\phantom{\rule{-2.84544pt}{0ex}}{I}_{a+}^{\alpha}{}^{RL}\phantom{\rule{-2.84544pt}{0ex}}{D}_{a+}^{\alpha}f(x)=f(x)-\sum _{j=1}^{n}\frac{{f}^{(\alpha -j)}(a)}{\mathrm{\Gamma}(\alpha -j+1)}{(x-a)}^{\alpha -j}$$
- If $f(x)\in {C}^{n}[a,b]$ or $f(x)\in A{C}^{n}[a,b]$, then$${}^{RL}\phantom{\rule{-2.84544pt}{0ex}}{I}_{a+}^{\alpha}{}^{C}\phantom{\rule{-2.84544pt}{0ex}}{D}_{a+}^{\alpha}f(x)=f(x)-\sum _{k=0}^{n-1}\frac{{f}^{(k)}(a)}{k!}{(x-a)}^{k}$$

#### 2.6. Grünwald-Letnikov Derivatives

#### 2.7. Liouville Derivatives

**Definition**

**2.**

## 3. Definite Fractional Integrals

#### 3.1. Integrals in ${\mathbb{R}}^{2}$ and ${\mathbb{R}}^{3}$

**Definition**

**3.**

**Definition**

**4.**

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

FD | Fractional derivative |

FI | Fractional integral |

RL | Riemann-Liouville |

L | Liouville |

C | Caputo |

GL | Grünwald-Letnikov |

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**MDPI and ACS Style**

Ortigueira, M.; Machado, J.
Fractional Definite Integral. *Fractal Fract.* **2017**, *1*, 2.
https://doi.org/10.3390/fractalfract1010002

**AMA Style**

Ortigueira M, Machado J.
Fractional Definite Integral. *Fractal and Fractional*. 2017; 1(1):2.
https://doi.org/10.3390/fractalfract1010002

**Chicago/Turabian Style**

Ortigueira, Manuel, and José Machado.
2017. "Fractional Definite Integral" *Fractal and Fractional* 1, no. 1: 2.
https://doi.org/10.3390/fractalfract1010002