Editing Fractional calculus (section)

== Historical notes ==
In [[applied mathematics]] and mathematical analysis, a '''fractional derivative''' is a derivative of any arbitrary order, real or complex. Its first appearance is in a letter written to [[Guillaume de l'Hôpital]] by [[Gottfried Wilhelm Leibniz]] in 1695.<ref name=Derivative>{{cite journal |last=Katugampola |first=Udita N. |date=15 October 2014 |title=A New Approach To Generalized Fractional Derivatives |url=https://www.emis.de/journals/BMAA/repository/docs/BMAA6-4-1.pdf |journal=Bulletin of Mathematical Analysis and Applications |volume=6 |issue=4 |pages=1–15 |arxiv=1106.0965 }}</ref> Around the same time, Leibniz wrote to [[Johann Bernoulli]] about derivatives of "general order".<ref name=":1">{{Cite book |last=Miller |first=Kenneth S. |title=An Introduction to the Fractional Calculus and Fractional Differential Equations |last2=Ross |first2=Bertram |date=1993 |publisher=Wiley |isbn=978-0-471-58884-9 |location=New York |pages=1–2}}</ref> In the correspondence between Leibniz and [[John Wallis]] in 1697, Wallis's infinite product for <math>\pi/2</math> is discussed. Leibniz suggested using differential calculus to achieve this result. Leibniz further used the notation <math>{d}^{1/2}{y}</math> to denote the derivative of order {{sfrac|1|2}}.<ref name=":1" /> 

Fractional calculus was introduced in one of [[Niels Henrik Abel]]'s early papers<ref>{{cite journal |title=Oplösning af et Par Opgaver ved Hjelp af bestemte Integraler (Solution de quelques problèmes à l'aide d'intégrales définies, Solution of a couple of problems by means of definite integrals) |year=1823 |journal=Magazin for Naturvidenskaberne |place=Kristiania (Oslo) |pages=55–68 |author=Niels Henrik Abel |url=https://abelprize.no/sites/default/files/2021-04/Magazin_for_Naturvidenskaberne_oplosning_av_et_par1_opt.pdf}}</ref> where all the elements can be found: the idea of fractional-order integration and differentiation, the mutually inverse relationship between them, the understanding that fractional-order differentiation and integration can be considered as the same generalized operation, and the unified notation for differentiation and integration of arbitrary real order.<ref>{{cite journal |doi=10.1515/fca-2017-0057 |title=Niels Henrik Abel and the birth of fractional calculus |year=2017 |journal=Fractional Calculus and Applied Analysis |pages=1068–1075 |last1=Podlubny |first1=Igor |last2=Magin |first2=Richard L. |last3=Trymorush |first3=Irina |volume=20 |issue=5 |arxiv=1802.05441 |s2cid=119664694}}</ref>
Independently, the foundations of the subject were laid by [[Liouville]] in a paper from 1832.<ref>{{Citation |last=Liouville |first=Joseph |author-link=Joseph Liouville |year=1832 |title=Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions |journal=Journal de l'École Polytechnique |volume=13 |pages=1–69 |location=Paris |url=https://gallica.bnf.fr/ark:/12148/bpt6k4336778/f2.item.r=Joseph%20Liouville}}.</ref><ref>{{Citation |last=Liouville |first=Joseph |author-link=Joseph Liouville |year=1832 |title=Mémoire sur le calcul des différentielles à indices quelconques |journal=Journal de l'École Polytechnique |volume=13 |pages=71–162 |location=Paris |url=https://gallica.bnf.fr/ark:/12148/bpt6k4336778/f72.image}}.</ref><ref>For the history of the subject, see the thesis (in French): Stéphane Dugowson, [http://s.dugowson.free.fr/recherche/dones/index.html ''Les différentielles métaphysiques''] (''histoire et philosophie de la généralisation de l'ordre de dérivation''), Thèse, Université Paris Nord (1994)</ref> [[Oliver Heaviside]] introduced the practical use of [[operational calculus|fractional differential operators]] in electrical transmission line analysis circa 1890.<ref>For a historical review of the subject up to the beginning of the 20th century, see: {{cite journal |doi=10.1016/0315-0860(77)90039-8 |title=The development of fractional calculus 1695–1900 |year=1977 |journal=Historia Mathematica |pages=75–89 |author=Bertram Ross |volume=4 |s2cid=122146887 |doi-access=}}</ref> The theory and applications of fractional calculus expanded greatly over the 19th and 20th centuries, and numerous contributors have given different definitions for fractional derivatives and integrals.<ref>{{cite journal |last1=Valério |first1=Duarte |last2=Machado |first2=José |last3=Kiryakova |first3=Virginia |author3-link=Virginia Kiryakova |date=2014-01-01 |title=Some pioneers of the applications of fractional calculus |journal=Fractional Calculus and Applied Analysis |volume=17 |issue=2 |pages=552–578 |doi=10.2478/s13540-014-0185-1 |hdl=10400.22/5491 |s2cid=121482200 |issn=1314-2224 |hdl-access=free}}</ref>