# Desiderata for Fractional Derivatives and Integrals

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## Abstract

**:**

- (a)
- Integrals ${\mathit{I}}_{}^{\alpha}$ and derivatives ${\mathit{D}}_{}^{\alpha}$ of fractional order $\alpha $ should be linear operators on linear spaces3.
- (b)
- On some subset4 ${\mathsf{G}}_{\left(\mathrm{b}\right)}\subset \mathsf{D}\left({\mathit{I}}_{}^{\alpha}\right)\cap {\mathit{I}}_{}^{\beta}\left[\mathsf{D}\left({\mathit{I}}_{}^{\beta}\right)\right]\cap \mathsf{D}\left({\mathit{I}}_{}^{\alpha +\beta}\right)\ne \varnothing $ the index law (semigroup property)$$\begin{array}{c}\hfill ({\mathit{I}}_{}^{\alpha}\circ {\mathit{I}}_{}^{\beta})f={\mathit{I}}_{}^{\alpha +\beta}f\end{array}$$
- (c)
- Restricted to a suitable subset ${\mathsf{G}}_{\left(\mathrm{c}\right)}\subset \mathsf{D}\left({\mathit{I}}_{}^{\alpha}\right)$ of the domain of ${\mathit{I}}_{}^{\alpha}$ the fractional derivatives ${\mathit{D}}_{}^{\alpha}$ of order $\alpha $ operate as left inverses$$\begin{array}{c}\hfill {\mathit{D}}_{}^{\alpha}\circ {\mathit{I}}_{}^{\alpha}={\mathsf{1}}_{{\mathsf{G}}_{\left(\mathrm{c}\right)}}\end{array}$$
- (d)
- There is a subset ${\mathsf{G}}_{\left(\mathrm{d}\right)}\subset \mathsf{D}\left({\mathit{D}}_{}^{\alpha}\right)$ of the domain of ${\mathit{D}}_{}^{\alpha}$ such that the limits$$\begin{array}{cc}\hfill {g}^{1}={\mathit{D}}_{}^{1}f& =\underset{\alpha \to 1}{lim}{\mathit{D}}_{}^{\alpha}f,\phantom{\rule{2.em}{0ex}}f\in {\mathsf{G}}_{\left(\mathrm{d}\right)},\hfill \end{array}$$$$\begin{array}{cc}\hfill {g}^{0}={\mathit{D}}_{}^{0}f& =\underset{\alpha \to 0}{lim}{\mathit{D}}_{}^{\alpha}f,\phantom{\rule{2.em}{0ex}}f\in {\mathsf{G}}_{\left(\mathrm{d}\right)},\hfill \end{array}$$
- (e)
- The limiting map ${\mathit{D}}_{}^{0}={\mathsf{1}}_{{\mathsf{G}}_{\left(\mathrm{d}\right)}}$ is the identity on ${\mathsf{G}}_{\left(\mathrm{d}\right)}$, i.e., ${g}^{0}=f$;
- (f)
- The limiting map ${\mathit{D}}_{}^{1}=\mathit{D}$ is a derivation on ${\mathsf{G}}_{\left(\mathrm{d}\right)}$. This means it is possible to define a multiplication $\xb7:{\mathsf{G}}_{\left(\mathrm{d}\right)}\times {\mathsf{G}}_{\left(\mathrm{d}\right)}\to {\mathsf{G}}_{\left(\mathrm{d}\right)}$ on ${\mathsf{G}}_{\left(\mathrm{d}\right)}$ such that the Leibniz rule$$\begin{array}{c}\hfill \mathit{D}(f\xb7g)=g\xb7\left(\mathit{D}f\right)+f\xb7\left(\mathit{D}g\right)\end{array}$$

## Appendix A

- (a)
- $T\left(0\right)={\mathsf{1}}_{\mathsf{X}}$.
- (b)
- $T\left(t\right)T\left(s\right)=T(t+s)$ for all $t,s\ge 0$.
- (c)
- For every $x\in \mathsf{X}$ the orbit maps ${y}_{x}:t\mapsto {y}_{x}\left(t\right):=T\left(t\right)x$ are continuous from $[0,\infty )$ into $\mathsf{X}$.

## References

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1 | properties to be desired. |

2 | It is common to use only one of the symbols $\mathit{I}$ or $\mathit{D}$ in the sense that either ${\mathit{D}}^{\alpha}={\mathit{I}}^{-\alpha}$ or ${\mathit{I}}^{\alpha}={\mathit{D}}^{-\alpha}$. In this paper we keep the distinction between $\mathit{I}$ and $\mathit{D}$ by assuming $\mathrm{Re}\alpha \ge 0$ unless otherwise specified. This entails discussing the case $\mathrm{Re}\alpha =0$ separately whenever necessary. |

3 | Dependencies of ${\mathit{I}}_{}^{\alpha}$ and ${\mathit{D}}_{}^{\alpha}$ on other parameters are usually present, but notationally suppressed. |

4 | Here the index (b) refers to desideratum (b). The same applies in desiderata (d)–(f) below. |

5 | (a) for all $f,g\in \mathsf{X}$ also $f+g\in \mathsf{X}$, (b) $f+g=g+f$, (c) $f+(g+h)=(f+g)+g$, (d) there exists an element $0\in \mathsf{X}$ (called origin ) such that $f+0=f$ for all $f\in \mathsf{X}$, (e) for all $f\in \mathsf{X}$ there is an element $-f\in \mathsf{X}$ such that $f+(-f)=0$ (f) for all $a\in \mathbb{R}$ (or $a\in \mathbb{C}$) and $f\in \mathsf{X}$ an element $af\in \mathsf{X}$ is defined, (g) for all $a\in \mathbb{R}$ (or $a\in \mathbb{C}$) and $f,g\in \mathsf{X}$ one has $a(f+g)=af+ag$, (h) for all $a,b\in \mathbb{R}$ (or $a,b\in \mathbb{C}$) and $f\in \mathsf{X}$ one has $(a+b)f=af+bf$, (i) for all $a,b\in \mathbb{R}$ (or $a,b\in \mathbb{C}$) and $f\in \mathsf{X}$ one has $a\left(bf\right)=\left(ab\right)f$, and (j) $1f=f$ for all $f\in \mathsf{X}$. |

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**MDPI and ACS Style**

Hilfer, R.; Luchko, Y. *Desiderata* for Fractional Derivatives and Integrals. *Mathematics* **2019**, *7*, 149.
https://doi.org/10.3390/math7020149

**AMA Style**

Hilfer R, Luchko Y. *Desiderata* for Fractional Derivatives and Integrals. *Mathematics*. 2019; 7(2):149.
https://doi.org/10.3390/math7020149

**Chicago/Turabian Style**

Hilfer, Rudolf, and Yuri Luchko. 2019. "*Desiderata* for Fractional Derivatives and Integrals" *Mathematics* 7, no. 2: 149.
https://doi.org/10.3390/math7020149