Series Representation of Power Function

Kolosov Petro (1)

1 Odessa National I.I.Mechnikov University

In this paper described numerical expansion of natural-valued power function $x^n$, in point $x=x_0$ where $n, \ x_0$ - natural numbers. Applying numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Binomial sum. Additionally, in section 4 exponential function’s $e^x$ representation is shown.

Mots clés: Newton's Binomial Theorem; Calculus; Maths; Mathematics; Math; Algebra; Mathematical analysis; Functional analysis; STEM; General Mathematics; Numercal methods; Classical Analysis and ODEs; Analysis of PDEs; Discrete Mathematics; q-derivative; Applied Mathematics; Backward Finite Difference; Jackson derivative; q-calculus; h-calculus; Quantum calculus; q-difference; Quantum algebra; Qunatum calculus; Hypergeometric series; Hypergeometric function; Time Scale Calculus; Power quantum calculus; Quantum difference; Quantum variatoinal calculus; h-difference; arXiv; Preprint; Open science; arXiv.org; Open access; 0000-0002-6544-8880; Series Expansion; Taylor's theorem; Taylor's formula; Taylor's series; Taylor's polynomial; Analytic function; Series representation; Polynomial expansion; Taylor theorem; Taylor formula; Taylor series; Taylor polynomial; Maclaurin Series; Petro Kolosov; Kolosov Petro; Kolosov; kolosov_p_1; KolosovP; petro-kolosov; kolosov-petro; petrokolosov; kolosov.petro; Kolosov_Petro; petro.kolosov.9; Monomial; Polynomial; Power function; Power (mathematics); Power series (mathematics); Exponential function; Exponentiation; Mathematical Series; Cube (Algebra); Euler number; Finite difference; Perfect cube; Diophantine equations; Divided difference; Ordinary differential equation; Derivative; High order finite difference; Partial difference; Calculus of variations; Partial differential equation; High order derivative; Partial derivative; Binomial coefficient; Differential calculus; Central Finite difference; Finite difference coefficient; Forward Finite Difference; Numerical Differentiation; Difference Equations; Numerical analysis; Science; Multinomial theorem; Binomial expansion; Pascal triangle; Binomial theorem; Newton's interpolation formula; Binomial Series; Pascal’s triangle; Differentiation; Derivatives; Finite differences; Number theory; Binomial Sum; Differential equations

Domaines: Mathématiques [math]; Théorie des nombres [math.NT]; Analyse classique [math.CA]; Systèmes dynamiques [math.DS]

Kolosov Petro. Series Representation of Power Function. 2017. ⟨hal-01283042v6⟩