IJPAM: Volume 40, No. 1 (2007)

UNIFIED FORMULAS FOR INTEGER AND FRACTIONAL
ORDER SYMBOLIC DERIVATIVES AND INTEGRALS
OF THE POWER-INVERSE TRIGONOMETRIC CLASS I

Mhenni M. Benghorbal
Department of Mathematics and Statistics
Concordia University
Montreal, QC, H3G 1M8, CANADA
e-mails: mbenghorbal@gmail.com, mhenni@mathstat.concordia.ca


Abstract.A complete solution to the problem of symbolic differentiation and integration of any order (integer, fractional, or real) of the power-inverse trigonometric classes has been given. In this work, we tackle the power-inverse tangential class
\begin{multline}
\bigg\{f(x):f(x)=\sum_{j=0}^{\ell}p_j(x^{\alpha_j}) \arctan\lef...
...slash \{0\}\,, \gamma_i \in \mathbb{R}\backslash
\{0\} \bigg\}\,,
\end{multline}
where $p_j$'s are polynomials of certain degrees. We give a unified formula for symbolic derivatives and integrals of any order. The approach does not depend on integration techniques. The formula, in general, is in terms of the $H$-function, but in many cases can be simplified for less general functions. Arbitrary (integer, fractional or real) order of differentiation is found according to the Riemann-Liouville definition, whereas we adopt the generalized Cauchy $n$-fold integral definition for arbitrary order of integration.

Received: August 7, 2007

AMS Subject Classification: 26A33

Key Words and Phrases: fractional derivatives, fractional integrals, $H$-function, $G$-function

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 40
Issue: 1