Differentiation of Arbitrary Order

Abstract

1. Out of Liouville’s early work on electromagnetism grew an interest in the theory known today under the name fractional calculus. It is a theory that generalizes the meaning of the differential and integral operators $\frac{{{d^\mu }}}{{d{x^\mu }}}$ from the well-known cases where μ is an integer ($\frac{{{d^{ - 1}}}}{{d{x^{ - 1}}}}$ meaning integration) to the cases when μ is a real rational or irrational or even complex number. Liouville’s own term, calcul des différentielles à indices quelconques (differential calculus of arbitrary index (or order)), is therefore more appropriate than the misleading “fractional calculus.” This was the first field in which Liouville published extensively and to which his name is still attached, in that the modern definition of differentiation of arbitrary order is called the Riemann-Liouville definition. He published 9 papers on the subject during the period from 1832 to 1837 and one late comer in 1855.