Means and non-real Intersection Points of Taylor Polynomials

TL;DR

This paper investigates the complex roots of differences between Taylor polynomials of functions, especially for the case r=3, revealing new means based on the real parts of these roots.

Contribution

It introduces a novel analysis of non-real roots of Taylor polynomial differences, defining new means via the roots' real parts, especially for polynomial functions.

Findings

For f(z)=z^p with p not in {0,1,2,3}, non-real roots have real parts between a and b.

A cubic polynomial g is constructed to determine the real part of complex roots.

The paper proves the existence of roots in (a,b) using the Intermediate Value Theorem.

Abstract

Suppose that f has continuous derivatives thru order r+1 for x>0, and let P_{c} denote the Taylor polynomial to f of order r at x=c,c>0. In a previous paper of the author, it was shown that if r is an odd whole number and the (r+1)st derivative of f is nonzero on [a,b], then there is a unique x_{0},a<x_{0}<b, such that P_{a}(x_{0})=P_{b}(x_{0}). This defines a mean, depending on f and r, given by m(a,b)=x_{0}. In this paper we discuss the real parts of the pairs of complex conjugate non-real roots of P_{b}-P_{a}. We prove some results for r in general, but our most significant results are for the case r=3. We prove in that case that if f(z)=z^{p}, where p is an integer, p not equal to 0,1,2, or 3, then P_{b}-P_{a} has non-real roots with real part strictly between a and b for any 0<a<b. This defines a countable family of means. We construct a cubic polynomial, g, whose real root gives the real part of the pair of complex conjugate non-real roots of P_{b}-P_{a}. Instead of working directly with a formula for the roots of a cubic, we use the Intermediate Value Theorem to show that g has a root in (a,b).