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Journal of Mathematics
Volume 2015, Article ID 134842, 8 pages
http://dx.doi.org/10.1155/2015/134842
Research Article

Exact Discrete Analogs of Derivatives of Integer Orders: Differences as Infinite Series

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia

Received 30 July 2015; Accepted 8 October 2015

Academic Editor: Mario Ohlberger

Copyright © 2015 Vasily E. Tarasov. 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

New differences of integer orders, which are connected with derivatives of integer orders not approximately, are proposed. These differences are represented by infinite series. A characteristic property of the suggested differences is that its Fourier series transforms have a power-law form. We demonstrate that the proposed differences of integer orders are directly connected with the derivatives . In contrast to the usual finite differences of integer orders, the suggested differences give the usual derivatives without approximation.