# IJPAM: Volume 40, No. 1 (2007)

**UNIFIED FORMULAS FOR INTEGER AND FRACTIONAL**

ORDER SYMBOLIC DERIVATIVES AND INTEGRALS

OF THE POWER-INVERSE TRIGONOMETRIC CLASS I

ORDER SYMBOLIC DERIVATIVES AND INTEGRALS

OF THE POWER-INVERSE TRIGONOMETRIC CLASS I

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*

where '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 -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 -fold integral definition for arbitrary order of
integration.

**Received: **August 7, 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:** 40

**Issue:** 1