Ramanujan常数的级数展开、性质及其应用

发布时间：2018-03-19 23:13

本文选题：Ramanujan常数　切入点：Beta函数　出处：《浙江理工大学》2017年硕士论文　论文类型：学位论文

【摘要】：众所周知,Gauss超几何函数F(a,b;c;x)在特殊函数中具有极为重要的地位,它与许多其他类型的特殊函数相关,其性质和Γ-函数,ψ-函数以及Beta函数B(a,b)密切相关。Ramanujan常数R(a)不仅在零平衡的Gauss超几何函数F(a,1-a;1;x)的研究中起着至关重要的作用,在特殊函数的一些其它领域也是必不可少的。例如,在对广义椭圆积分κa(r)和εa(r),广义模方程的解φK(a,r)以及由φK(a,r)定义的λ(a,K)和ηK(a,t)等特殊函数分析性质的研究中经常用到Ramanujan常数R(a)。但R(a)的已知性质尚不能满足应用中的需要,而揭示R(a)性质的主要障碍之一是缺乏行之有效的研究工具。不少研究工作表明,R(a)的级数展开是重要而有效的研究工具。本文的主要目的是建立Ramanujan常数R(a)的不同类型级数、进一步揭示Ramanujan常数与Beta函数的紧密关系,并通过研究R(a)与一些初等函数组合的性质,获得R(a)的一些重要性质。本文由以下三章构成:第一章,主要介绍了本文的研究背景,并引入本文所涉及的一些概念、记号和部分已有结果。第二章,我们首先建立了 Ramanujan常数R(a)、B(a)= B(a,1-a)的不同类型的级数展开式和R(a)-B(a)的幂级数展开式,并运用这些结果得出了 Rieman zeta函数满足的一些等式。第三章,通过研究Ramanujan常数R(a)与某些初等函数组合的分析性质,获得了 R(a)的一些渐近精确的不等式,并改进了某些已有结果。
[Abstract]:It is well known that the Gauss hypergeometric function (FG) plays a very important role in special functions, and it is related to many other special functions. Its properties are closely related to 螕-functions, 蠄-functions and Beta functions. Ramanujan constant Rao _ a) not only plays an important role in the study of Gauss hypergeometric functions with zero equilibrium, but also in some other fields of special functions. It is often used in the study of the analytical properties of some special functions such as the generalized elliptic integral 魏 ~ (a) and 蔚 ~ (a) ~ (r), the solution of the generalized mode equation 蠁 K ~ (a) ~ r), and the definition of 位 ~ (a) ~ n ~ (K) and 畏 ~ K ~ (a ~ (t)). However, the known properties of the Ramanujan constant Rao _ (a) can not meet the needs of application. One of the main obstacles to reveal the properties of Ria is the lack of effective research tools. Many researches show that the series expansion of Ria) is an important and effective research tool. The main purpose of this paper is to establish different types of series of Ramanujan constant. The close relationship between the Ramanujan constant and the Beta function is further revealed, and some important properties are obtained by studying the properties of RIA) and some elementary functions. This paper is composed of the following three chapters: in Chapter 1, the research background of this paper is introduced. In chapter 2, we first establish different types of series expansions and power series expansions of the Ramanujan constant R ~ (n) ~ (a) ~ (1 ~ (a)) and the power series expansions of R _ (a) ~ (B ~ (1) ~ (a)), and then introduce some concepts, notations and some known results of this paper. In chapter two, we first establish the different types of series expansions of the Ramanujan constant, By using these results, some equations of Rieman zeta functions are obtained. In Chapter 3, by studying the analytical properties of the combination of Ramanujan constant Ru (a) and some elementary functions, we obtain some asymptotically exact inequalities. Some existing results are improved.
【学位授予单位】：浙江理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.6

【参考文献】