Taylor多项式的均值与非实交点
Means and non-real Intersection Points of Taylor Polynomials
摘要：
假设f具有通过r+1阶的连续导数，其中x>0，并且令P_{c}表示f在x=c，c>0处的r阶泰勒多项式。在作者的前一篇文章中，证明了如果r是整数且f在[a，b]上的（r+1）阶导数不为零，则存在唯一的x_{0}，a< x_{0} <b，使得P_{a}（x_{0}）= P_{b}（x_{0}）.这定义了一个均值，取决于f和r，由m（a，b）= x_{0}给出。本文讨论了P_{b} - P_{a}的复共轭的非实根对的实部.我们证明了一些结果r一般，但我们最显着的结果的情况下r=3。我们证明了在这种情况下，如果f（z）=z^{p}，其中p是一个整数，p不等于0，1，2或3，则P_{b} - P_{a}有非实根，其实部严格在a和b之间，对任意0<a<b.这定义了一个可数的均值族。我们构造了一个三次多项式g，它的实根给出了P_{b} - P_{a}的复共轭的非实根对的实部.我们不直接用三次函数的根的公式，而是用中值定理证明g在（a，b）中有根。
英文摘要：
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).
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