Gamma函数的对数完全单调性质及其相关性质的研究

发布时间：2018-01-15 03:02

本文关键词：Gamma函数的对数完全单调性质及其相关性质的研究　出处：《西北大学》2015年硕士论文　论文类型：学位论文

【摘要】：通过数百年来对Gamma函数的系统研究,揭示了它具有一系列独特的性质,关于Gamma函数的问题,是数学领域内的一个十分有意义的课题.本研究在已有结论的基础上,深入地研究了Gamma函数、Psi函数以及Gamma之商函数的一些性质,本文主要做了如下工作：1.研究了函数G(x),H(x)的对数完全单调性以及函数ν(x)的单调性,得出一些关于Γ(x+1)的双向不等式,由此给出了一些关于n!的渐近表达式；通过研究函数fα,β(x)的对数完全单调性,获得了Gamma函数与指数平均数之间的关系.2.根据Gamma之商函数的性质,得出了Gamma函数之商的不等式,通过函数的一些特殊性质改进了Keckic — Vasic双向不等式.3.通过对Psi函数部分性质的研究,证实了一些简单的不等式,并在新的方法指导下证明了部分已有不等式成立.
[Abstract]:Through the systematic research on Gamma function for hundreds of years, it is revealed that it has a series of unique properties, about the problem of Gamma function. It is a very meaningful subject in the field of mathematics. On the basis of the existing conclusions, the properties of the Gamma function PSI function and the quotient function of Gamma are studied in depth. The main work of this paper is as follows: 1. We study the logarithmic complete monotonicity of the function G _ (x) and the monotonicity of the function 谓 _ (x), and obtain some bidirectional inequalities about 螕 ~ (X _ 1). This gives us some information about n! The asymptotic expression of; By studying the logarithmic monotonicity of functions f 伪, 尾 X, the relationship between Gamma function and exponential average is obtained. 2. According to the properties of quotient function of Gamma. The inequality of quotient of Gamma function is obtained, and the Keckic-Vasic bidirectional inequality. 3. The partial properties of Psi function are improved by some special properties of Keckic-Vasic function. Some simple inequalities are proved and some existing inequalities are proved under the guidance of new methods.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O174

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