IJPAM: Volume 37, No. 3 (2007)

A UNIFIED FORMULA FOR ARBITRARY ORDER
SYMBOLIC DERIVATIVES AND INTEGRALS OF
THE POWER-EXPONENTIAL CLASS

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


Abstract.We give a complete solution to the problem of symbolic differentiation and integration of arbitrary (integer, fractional, or real) order of the power-exponential class

\begin{displaymath} \left\{f(x):f(x)=\sum_{j=1}^{\ell}p_j(x^{\alpha_j})e^{\beta_... ...ash \{0\}, \gamma_j \in \mathbb{R}\backslash \{0\} \right\}\,, \end{displaymath} (1)

through a unified formula in terms of the $H$-function which can, in many cases, be simplified to less general functions. We begin our talk by discussing a less general class of functions given by
\begin{displaymath} \left\{f(x):f(x)=\sum_{j=1}^{\ell}p_j(x)e^{\beta_j x}\,, \beta_j \in \mathbb{C} \right\}\,, \end{displaymath} (2)

which is a subclass of the power-exponential class. It has the property that its $n$th derivative and integral formulas of integer order belongs to the same class.

In Maple, the formulas correspond to invoking the commands $\mathrm{diff}(f(x)$, $x\$q)$ for differentiation and $\mathrm{int}(f(x), x\$q)$ for integration, where $q$ is an integer, a fraction, a real, or a symbol. They enhance the ability of computer algebra systems for computing derivatives and integrals of arbitrary orders at a point $x$.

The arbitrary order of differentiation is found according to the Riemann-Liouville definition, whereas the generalized Cauchy $n$-fold integral is adopted for arbitrary order of integration.

One of the key points in this work is that the approach does not depend on integration techniques.

Received: March 26, 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: 37
Issue: 3