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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 310679, 11 pages
http://dx.doi.org/10.1155/2013/310679
Research Article

-Dimensional Fractional Lagrange's Inversion Theorem

1Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah, Saudi Arabia
2Department of Astronomy, Faculty of Science, Cairo University, Cairo 12613, Egypt

Received 10 November 2012; Revised 14 January 2013; Accepted 20 January 2013

Academic Editor: Ciprian A. Tudor

Copyright © 2013 F. A. Abd El-Salam. 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

Using Riemann-Liouville fractional differential operator, a fractional extension of the Lagrange inversion theorem and related formulas are developed. The required basic definitions, lemmas, and theorems in the fractional calculus are presented. A fractional form of Lagrange's expansion for one implicitly defined independent variable is obtained. Then, a fractional version of Lagrange's expansion in more than one unknown function is generalized. For extending the treatment in higher dimensions, some relevant vectors and tensors definitions and notations are presented. A fractional Taylor expansion of a function of -dimensional polyadics is derived. A fractional -dimensional Lagrange inversion theorem is proved.