本文選題：Ramanujan常數(shù)　切入點(diǎn)：Beta函數(shù)　出處：《浙江理工大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：眾所周知,Gauss超幾何函數(shù)F(a,b;c;x)在特殊函數(shù)中具有極為重要的地位,它與許多其他類型的特殊函數(shù)相關(guān),其性質(zhì)和Γ-函數(shù),ψ-函數(shù)以及Beta函數(shù)B(a,b)密切相關(guān)。Ramanujan常數(shù)R(a)不僅在零平衡的Gauss超幾何函數(shù)F(a,1-a;1;x)的研究中起著至關(guān)重要的作用,在特殊函數(shù)的一些其它領(lǐng)域也是必不可少的。例如,在對(duì)廣義橢圓積分κa(r)和εa(r),廣義模方程的解φK(a,r)以及由φK(a,r)定義的λ(a,K)和ηK(a,t)等特殊函數(shù)分析性質(zhì)的研究中經(jīng)常用到Ramanujan常數(shù)R(a)。但R(a)的已知性質(zhì)尚不能滿足應(yīng)用中的需要,而揭示R(a)性質(zhì)的主要障礙之一是缺乏行之有效的研究工具。不少研究工作表明,R(a)的級(jí)數(shù)展開(kāi)是重要而有效的研究工具。本文的主要目的是建立Ramanujan常數(shù)R(a)的不同類型級(jí)數(shù)、進(jìn)一步揭示Ramanujan常數(shù)與Beta函數(shù)的緊密關(guān)系,并通過(guò)研究R(a)與一些初等函數(shù)組合的性質(zhì),獲得R(a)的一些重要性質(zhì)。本文由以下三章構(gòu)成:第一章,主要介紹了本文的研究背景,并引入本文所涉及的一些概念、記號(hào)和部分已有結(jié)果。第二章,我們首先建立了 Ramanujan常數(shù)R(a)、B(a)= B(a,1-a)的不同類型的級(jí)數(shù)展開(kāi)式和R(a)-B(a)的冪級(jí)數(shù)展開(kāi)式,并運(yùn)用這些結(jié)果得出了 Rieman zeta函數(shù)滿足的一些等式。第三章,通過(guò)研究Ramanujan常數(shù)R(a)與某些初等函數(shù)組合的分析性質(zhì),獲得了 R(a)的一些漸近精確的不等式,并改進(jìn)了某些已有結(jié)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.
【學(xué)位授予單位】：浙江理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O174.6

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本文編號(hào)：1636451