# IJPAM: Volume 37, No. 3 (2007)

**A UNIFIED FORMULA FOR ARBITRARY ORDER**

SYMBOLIC DERIVATIVES AND INTEGRALS OF

THE POWER-EXPONENTIAL CLASS

SYMBOLIC DERIVATIVES AND INTEGRALS OF

THE POWER-EXPONENTIAL CLASS

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*

which is a subclass of the

*power-exponential class*. It has the property that its th derivative and integral formulas of integer order belongs to the same class.

In *Maple*, the formulas correspond to invoking the commands
, for differentiation and
for integration, where 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 .

The arbitrary order of differentiation is found according to the Riemann-Liouville definition, whereas the generalized Cauchy -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, -function, -function

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2007

**Volume:** 37

**Issue:** 3