Various approaches to fractional derivation

The question of fractional derivatives was addressed as early as 1695 by Leibnitz in a letter to de L'Hospital, but when de L'Hospital asked him what the half-order derivative of the function x could be, Leibnitz replied that this led to a paradox from which useful consequences would one day be drawn. More than 300 years later, we're only just beginning to overcome the difficulties. Numerous mathematicians have addressed this question, in particular Euler (1730), Fourier (1822), Abel (1823), Liouville (1832), Riemann (1847) and others. Various approaches have been used to generalize the notion of derivation to non-integer orders, including :

• the limit of the rate of increase of a function, generalized in the form of the Grünwald-Letnikov formula, which is very useful numerically;

• integration, the inverse operation,...