發(fā)布時(shí)間：2018-01-15 03:02

  本文關(guān)鍵詞：Gamma函數(shù)的對數(shù)完全單調(diào)性質(zhì)及其相關(guān)性質(zhì)的研究　出處：《西北大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： Gamma函數(shù) Psi函數(shù) 對數(shù)完全單調(diào) 完全單調(diào) Gamma之商函數(shù) 不等式

【摘要】：通過數(shù)百年來對Gamma函數(shù)的系統(tǒng)研究,揭示了它具有一系列獨(dú)特的性質(zhì),關(guān)于Gamma函數(shù)的問題,是數(shù)學(xué)領(lǐng)域內(nèi)的一個(gè)十分有意義的課題.本研究在已有結(jié)論的基礎(chǔ)上,深入地研究了Gamma函數(shù)、Psi函數(shù)以及Gamma之商函數(shù)的一些性質(zhì),本文主要做了如下工作：1.研究了函數(shù)G(x),H(x)的對數(shù)完全單調(diào)性以及函數(shù)ν(x)的單調(diào)性,得出一些關(guān)于Γ(x+1)的雙向不等式,由此給出了一些關(guān)于n!的漸近表達(dá)式；通過研究函數(shù)fα,β(x)的對數(shù)完全單調(diào)性,獲得了Gamma函數(shù)與指數(shù)平均數(shù)之間的關(guān)系.2.根據(jù)Gamma之商函數(shù)的性質(zhì),得出了Gamma函數(shù)之商的不等式,通過函數(shù)的一些特殊性質(zhì)改進(jìn)了Keckic — Vasic雙向不等式.3.通過對Psi函數(shù)部分性質(zhì)的研究,證實(shí)了一些簡單的不等式,并在新的方法指導(dǎo)下證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 謂 _ (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 偽, 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.
【學(xué)位授予單位】：西北大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O174

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本文編號：1426467