Unified formulas for arbitrary order symbolic derivatives and anti-derivatives of the power-inverse hyperbolic class 1

Abstract

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 less general functions. In this work, we consider two subclasses of the power-inverse hyperbolic class. Namely, the power-inverse hyperbolic sine class {f(x) : f(x) = Σ^l_j=1Pj(x^αj)arcsinh(β_jx^γj), α_j ∈ C, β_j ∈ C\{0},γ_j ∈ R\{0}, (1) and the power-inverse hyperbolic cosine class {f(x) : f(x) = Σ^l_j=1Pj(x^αj)arccosh(β_jx^γj), α_j ∈ C, β_j ∈ C\{0},γ_j ∈ R\{0}, (2) where pj's are polynomials of certain degrees. One of the key points in this work is that the approach does not depend on integration techniques 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.

The motivation of this work comes from the area of symbolic computation. The idea is that: Given a function f in a variable x, can CAS find a formula for the nth derivative, the nth anti-derivative, or both of f? This enhances the power of integration and differentiation of CAS. In Maple, the formulas correspond to invoking the commands diff(f(x) for the nth derivative and int(f(x), x$n) for the nth anti-derivative. A software exhibition will be given using Maple.

Example: A unified formula for arcsinh(√x) in terms of the Meijer G-function (arcsinh(√x)) ⁽ⁿ⁾ = x^(1/2--nover2√π G^1,2over_1,2 (1/2,1/2over0,n--1/2│x) , │x│ < 1. (3).

The above G-function reduces to the original function if n = 0. It gives derivatives of any order if n > 0 and anti-derivatives of any order if n < 0.

References

1. Mhenni Benghorbal, Power series solutions of fractional differential equations and symbolic derivatives and integrals, Ph.D. thesis, London, 2004.
2. Mhenni Benghorbal, A unified formula for arbitrary order symbolic derivatives and integrals of the power-exponentialclass, International Journal of Pure and Applied Mathematics (IJPAM) vol.37 (2007), no. 3, 349--365.
3. Mhenni Benghorbal, Unified formulas for integer and fractional order symbolic derivatives and integrals of the power-inverse trigonometric class I, Int. J. P. App. Math.(2007).
4. Mhenni Benghorbal, Unified formulas for integer and fractional order symbolic derivatives and integrals of the power-inverse trigonometric class II, To appear: Int. J. P. App. Math.(2008).

Index Terms

• Mathematics of computing

  • Mathematical analysis

    • Differential equations

      • Partial differential equations

    • Integral equations

    • Numerical analysis

      • Numerical differentiation

• Theory of computation

  • Design and analysis of algorithms