Unified Formulas For Arbitrary Order Symbolic Derivatives and Anti-Derivatives ofThe Power-Inverse Hyperbolic Class I

摘要

We continue on tackling and giving a complete solution to the problem of finding the nth derivative and the nth anti-derivative, where n can be an integer, a fraction, a real, or a symbol, of elementary and special classes of functions. In general,the solutions are given through unified formulas in terms of the Fox H-function which in many cases can be simplified to lessgeneral functions. In this work, we consider two subclasses of the power-inverse hyperbolic class. Namely, the power-inversehyperbolic sine class where pi's are polynomials of certain degrees. One of the key points in this work is that the approach does not depend onintegration techniques The arbitrary order of differentiation is found according to the Riemann-Liouville definition, whereasthe generalized Cauchy n-fold integral is adopted for arbitrary order of integration. The motivation of this work comes from the area of symbolic computation. The idea is that: Given a function f in a variablex, can CAS find a formula for the nth derivative, the nth anti-derivative, or both of f? This enhances the power of integrationand differentiation of CAS. In Maple, the formulas correspond to invoking the commands diff( f (x), x