Some fractional integral and derivative formulas revisited
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913539898212352 |
|---|---|
| author | Gonzales-Santander, Juan Luis Mainardi, Francesco |
| author_facet | Gonzales-Santander, Juan Luis Mainardi, Francesco |
| contents | In the most common literature about fractional calculus, we find that $_{a}D_{t}^{α}f\left( t\right) =\,_{a}I_{t}^{-α}f\left( t\right) $ is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of $_{a}I_{t}^{α}f\left( t\right) $ and $_{a}D_{t}^{α}f\left( t\right) $. In this sense, we prove that $_{0}D_{t}^{α}f\left( t\right) =\,_{0}I_{t}^{-α}f\left( t\right) $ is true for $f\left( t\right) =t^{ν-1}\log t$, and $f\left( t\right) =e^{λt}$, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for $_{-\infty }D_{t}^{α}\left\vert t\right\vert ^{-δ}$ and $_{-\infty }I_{t}^{α}\left\vert t\right\vert ^{-δ}$ found in the literature are incorrect; thus, we derive the correct ones, proving in turn that $_{-\infty }D_{t}^{α}\left\vert t\right\vert ^{-δ}=\,_{-\infty }I_{t}^{-α}\left\vert t\right\vert ^{-δ}$ holds true |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07181 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some fractional integral and derivative formulas revisited Gonzales-Santander, Juan Luis Mainardi, Francesco General Mathematics 26A33 In the most common literature about fractional calculus, we find that $_{a}D_{t}^{α}f\left( t\right) =\,_{a}I_{t}^{-α}f\left( t\right) $ is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of $_{a}I_{t}^{α}f\left( t\right) $ and $_{a}D_{t}^{α}f\left( t\right) $. In this sense, we prove that $_{0}D_{t}^{α}f\left( t\right) =\,_{0}I_{t}^{-α}f\left( t\right) $ is true for $f\left( t\right) =t^{ν-1}\log t$, and $f\left( t\right) =e^{λt}$, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for $_{-\infty }D_{t}^{α}\left\vert t\right\vert ^{-δ}$ and $_{-\infty }I_{t}^{α}\left\vert t\right\vert ^{-δ}$ found in the literature are incorrect; thus, we derive the correct ones, proving in turn that $_{-\infty }D_{t}^{α}\left\vert t\right\vert ^{-δ}=\,_{-\infty }I_{t}^{-α}\left\vert t\right\vert ^{-δ}$ holds true |
| title | Some fractional integral and derivative formulas revisited |
| topic | General Mathematics 26A33 |
| url | https://arxiv.org/abs/2410.07181 |