The development of fractional calculus 1695-1900
This paper describes an example of mathematical growth from scholarly curiosity to application. When Leibniz invented the symbol \(d^ny/dx^n\), L'Hospital in 1695 asked him what should happen if \(n\) be \(1/2\)? That is why it is called fractional calculus. But the subject is better called ``differentiation and Integration of arbitrary order'' because it was learned later that n could be any arbitrary number, real or complex. This paper discusses the first published fractional calculus by Lacroix in 1819 which was merely an exercise generalizing a case of integer order. Then the work of Liouville is discussed. He used a different method and was successful in applying his definition to problems in potential theory. But Peacock supported Lacroix's method while others supported Liouville's method. The controversy lasted almost a half century and was resolved by H. Laurent in 1888 [JFM 20.0270.01] by an analysis in the complex domain. The result was as De Morgan had predicted in 1840. Both methods were parts of a more general system. Laurent showed that integration of arbitrary order was the same as Riemann's definition which was published posthumously, namely \({}_0D_x^\nu f(x) = \frac 1{\Gamma(\nu)} \int_0^x (x-t)^{\nu - 1} f(t)dt\) but without a complementary function. Peacock made some serious errors in generalizing from a case of integer order. Liouville also made errors and Cayley remarked that Riemann's complementary function was indeterminate. So long after \textit{H. Laurent}' s work in 1888, there was a distrust of fractional operations. Thus, when Heaviside introduced his fractional operators in electrical circuit theory he was not received very well by the mathematicians of the period. This paper gives some elementary examples but is not as expository as it would be because the paper was written for a historical journal. The paper does have a modest bibliography. This paper was an outgrowth of part of a dissertation under the joint supervision of Prof. William Zlot, New York University (math. ed.) and Prof. Melvin Hausner, Courant Institute, New York University.
scientific article; zbMATH DE number 3557714