International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 5, Pages 315-325
doi:10.1155/S0161171203206050
New approach to the fractional derivatives
Institute of Mathematics, St. Cyril and Methodius University, P.O. Box 162, Skopje 1000, Macedonia
Received 15 June 2002
Copyright © 2003 Kostadin Trenčevski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We introduce a new approach to the fractional derivatives of the
analytical functions using the Taylor series of the functions. In
order to calculate the fractional derivatives of f, it is not
sufficient to know the Taylor expansion of f, but we should
also know the constants of all consecutive integrations of f.
For example, any fractional derivative of ex is ex only if
we assume that the nth consecutive integral of ex is ex
for each positive integer n. The method of calculating the
fractional derivatives very often requires a summation of
divergent series, and thus, in this note, we first introduce a
method of such summation of series via analytical continuation of
functions.