A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
type: Item
instance of: scholarly article
title: A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
zbMATH Open document ID: 1547.41026
author: Yan-Fang Li
author: Feng Qi
published in: Open Mathematics
publication date: 2024-09-26T00:00:00Z
review text: The authors pose themselves two problems:\N\N\textbf{1.} What is the MacLaurin expansionn of the even function\N\[\NF(x)=\begin{cases} \ln{\frac{2(1-\cos{x})}{x^2}}, &0<|x|< 2 \pi \\\N0, &x=0 \end{cases}\N\]\N\N\textbf{2.} Is the even function\N\[\NR(x)=\begin{cases} \frac{\ln{\frac{2(1-\cos{x})}{x^2}}}{\ln{\cos{x}}}, &0<|x|<\frac{\pi}{2} \\\N\frac{1}{6}, &x=0 \\\N0, &x=\pm \frac{\pi}{2} \end{cases}\N\]\Ndecreasing on the interval, $[0,\frac{\pi}{2}]$?\N\NThe answers are given in the paper and will be indicated below.\N\NThe layout of the paper is as follows\N\N\textbf{\S1. Motivation} ($1$ page)\N\N\textbf{\S2. Lemmas} ($1\frac{1}{2}$ page)\N\N\textbf{\S3. MacLaurin power series expansion} ($4$ pages)\N\NThe series is given (\textbf{Theorem 1}) in the form\N\[\NF(x)=-\sum_{n=1}^{\infty}\,\frac{E_{2n}}{(2n)!},\ |x|<2\pi,\N\]\Nwhere the numbers $E_{2n}$ are given explicitly as the values of intricate determnants.\N\N\textbf{\S4. Decreasing property} ($4\frac{1}{2}$ pages)\N\NThis is \textbf{Theorem 2}:\N\NThe function $R(x)$ is decreasingly maps $[0,\frac{\pi}{2}]$ onto $[0,\frac{1}{6}]$.\N\N\textbf{\S5. Conclusion} ($\frac{1}{2}$ page)\N\N\textbf{References} (16 items)
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 41A58
Mathematics Subject Classification ID: 26A09
Mathematics Subject Classification ID: 33B10
zbMATH DE Number: 7919412
zbMATH Keywords: Maclaurin power series expansion
zbMATH Keywords: decreasing property
zbMATH Keywords: logarithmic expression
zbMATH Keywords: cosine function
zbMATH Keywords: derivative formula
zbMATH Keywords: ratio of two differentiable functions
zbMATH Keywords: monotonicity rule
MaRDI profile type: Publication
cites work: Q4880419
cites work: A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers
cites work: Q4430468
cites work: Q5045847
cites work: Q4339229
cites work: Two closed forms for the Bernoulli polynomials
cites work: Some determinantal expressions and recurrence relations of the Bernoulli polynomials
cites work: Series expansions for real powers of sinc function and closed-form expression for specific partial Bell polynomials
DOI: 10.1515/MATH-2023-0159
description: scientific article; zbMATH DE number 7919412