# Fractional Calculus: differentiation and integration of non-integer order

## Marcus Webb

In 1695, Leibniz and L’Hôpital were discussing the newly developed differential calculus by letter. Considering the notation \(d^n/dx^n\), L’Hôpital asked ‘… and what if \(n\) be \(1/2\)?’. Leibniz replied: ‘It will lead to a paradox, from which one day useful consequences will be drawn.’ This was the birth of the fractional calculus. The idea is to generalise the notion of differentiation and integration of order \(1, 2, 3\) etc. to that of fractional order \(s\), i.e. where \(s\) is a real number. We look at the classic theory by authors such as Euler, Liouville, Riemann, and Riesz, then show how in the last 40 years real world phenomena have been modelled using the fractional calculus, confirming Leibniz’s prophecy. It is a very active research area with difficulties still unresolved.