Simplifying Polynomial Functions: an Analytic Expansion Approach

Mansour Khalid Hammad

doi:10.15866/iremos.v16i2.23432

Simplifying Polynomial Functions: an Analytic Expansion Approach

Mansour Khalid Hammad^(1*)

^(*) Corresponding author

Authors' affiliations

DOI: https://doi.org/10.15866/iremos.v16i2.23432

Abstract

This article aims to provide a comprehensive analysis of the Taylor, Maclaurin, Lagrange, and other power series. A significant aspect is the exploration of Mansour's Expansion, a generalization of the work of Taylor, Maclaurin, and others in the realm of power series. This paper demonstrates how Taylor and Maclaurin's Expansion can be viewed as a special case of this new Expansion, which is an intriguing concept. Establishing this relationship provides a fresh perspective on these classical power series and their applications. In addition to exploring the relationships between classical power series, this article also highlights the practical applications of Mansour's Expansion. One key advantage of this new Expansion is its efficient algorithm, which quickly determines whether a polynomial is a composition. Furthermore, the algorithm can determine if a given polynomial g is a function in f. These capabilities carry significant implications for solving and simplifying polynomial functions into lower-degree functions. Mansour’s Expansion provides a promising avenue for developing more efficient and effective methods for solving complex mathematical problems. It also provides a unique technique for solving an equation of the form f(x) = 0 not only by algebra, but also by iteration. It is an intriguing generalization of the work of Newton’s Method. This makes it an important area of research for mathematicians and scientists alike.
Copyright © 2023 Praise Worthy Prize - All rights reserved.

Keywords

Calculus; Differential Equations; Numerical Method

Full Text:

PDF

References

J. Stewart, D. Clegg, and S.Watson, S, Calculus: Early transcendental (9th ed.). (Cengage Learning, 2021).

B.Schneider, I. Miller, and B. V Saunders, Physics Today, 71(2), (NIST's Digital Library of Mathematical Functions, 2018).
https://doi.org/10.1063/PT.3.3846

S.S. Rattan, Mathematical methods: For science and engineering. (Pearson Education India, (2009).

Hammad, M., If f(x) is a Function in g(x), what are the Coefficients of that Function? A New Expansion for Analytic Function in Standard Form f(x) = Σk=0∞ Sk (g(x) - M)k, (2013) International Journal on Numerical and Analytical Methods in Engineering (IRENA), 1 (2), pp. 105-109.

R. Courant, F. John,. Introduction to Calculus and Analysis (Vol. 1). (Springer Science & Business Media,1999).
https://doi.org/10.1007/978-3-642-58604-0_1

E. Kreyszig, Advanced engineering mathematics (10th ed). (John Wiley & Sons, 2011).

R.L. Burden and , J. Douglas Faires Numerical Analysis. (8th Ed.), (Thomson Brooks/Cole, Belmont , 2005).

S.C. Chapra,and R.P. Canale, Numerical methods for engineers (7th ed.). (McGraw-Hill Education, New York, 2015).

S. P. Thompson, Calculus Made Easy.( St. Martin's Press, Newyork ,1998).
https://doi.org/10.1007/978-1-349-15058-8

J. D. Lambert, Numerical methods for ordinary differential equations (3rd ed.). (Springer, New York, 2013).

M. R. Osborne, Numerical methods for differential equations and optimization (2nd ed.). (Wiley, New York, 2010).

G.B. Thomas Jr. , M.D. Weir, and Hass, J. Thomas' Calculus. (12th Ed.) (Pearson, Boston, New York, 2010).

T.J. Ypma, Historical Development of the Newton-Raphson Method. SIAM Review. (SIAM Review, 1995).
https://doi.org/10.1137/1037125

V.V. Piterbarg, L.B.G Andersen, Interest Rate Modeling, Products and Risk Management, volume III. (Atlantic Financial Press, 2010).

Refbacks

There are currently no refbacks.

Username
Password
Remember me