A relevant part of the history of Joseph Liouville Bernhard Riemann 1826-1866 fractional calculus 1809-1882 In a paper written when Leibniz first introduced the began with the Liouville formally just a student Riemann, idea of a symbolic method papers of Abel and extended the that was published only and used the symbol Liouville. formula for the ten years after is death, dny/dxn = Dny In a letter to L`Hôpital in 1695 Leibniz raised the following derivative of he arrived to an Niels Abel n ax for the nth derivative, question: "Can the meaning of derivatives with integer order integral orderD e expression for fractional 1802-1829 where n is a non-negative be generalized to derivatives with non-integer orders?" to derivatives of integration that became integer. L`Hôpital was somewhat curious about that question and arbitrary order α one of the main formulae replied by another question to Leibniz: "What if the order will Dαeax = aαeax together with Liouville be 1/2?" construction. Leibniz in a letter dated September 30, 1695 - the exact Gottfried Leibniz birthday of the fractional calculus! — replied: "It will lead to a 1646 -1716 paradox, from which one day useful consequences will be Anton Grünwald Aleksey Letnikov drawn." 1838-1920 1837-1888 Hermann Weyl John Wallis Karl Weierstrass 1885-1955 1616-1703 Guillaume de l'Hôpital Grünwald and 1815-1897 Letnikov developed 1661- 1704 Leonhard Euler Harold Davis an approach to 1892-1974 1707-1783 fractional Historically, differentiation based Isaac Newton on the limit of a sum and Gottfried Leibniz independently Isaac Newton Euler observed that the result of the evaluation Magnus Mittag-Leffler discovered 1643-1727 n n p Godfrey Hardy calculus in the of d y/dx of the power function x has a 1846-1927 1877-1947 meaning for non-integer p. John Littlewood 17th century. 1885-1977 Marcel Riesz Pierre-Simon Laplace Laplace proposed the idea of 1886-1969 William Feller 1749-1827 Ian Sneddon differentiation of non-integer order for 1906-1970 functions representable by an integral 1919-2000 ∫T(t)t─xdt Joseph-Louis Lagrange Salvatore Pincherle 1736-1813 Hjalmar Holmgren 1853-1936 1822-1885 Andrew Gemant 1895 - 1983 Yury Rabotnov History of 1914-1985 Eric Love Mkhtar Djrbashjan 1912-2001 1918-1994 Fractional Calculus Joseph Fourier Paul Lévy 1768-1830 Fourier suggested the idea 1886-1971 2010 of using his integral representation of f(x) to define the derivative for J. Tenreiro Machado, Virginia Kiryakova, Francesco Mainardi Pavel Nekrasov non-integer order. 1853-1924 Hermann Kober Anatoly Kilbas 1948-2010 1888-1973 Arthur Erdélyi Antoni Zygmund 1908-1977 1900-1992 Jacques Hadamard 1865-1963 Oliver Heaviside © Copyright 1850-1925 J. Machado, V. Kiryakova, F. Mainardi 1700 1750 1800 1850 1900 1950.