Γ-函数的性质及其应用

论文编号:XXLW003 论文字数:8323,页数:38

摘 要

 经典数学中的函数，从Euler开始系统地研究至今已有二百多年了，它作为一种超越函数，具备了丰富和优美的特性，在数学的许多分支以及物理、工程等学科中都起着重要的作用。
 文中主要探讨：1、函数的性质研究；2、通过实例来说明，换元或分部积分若干次后化为函数，以及函数与函数、三角函数的关系；3、概率统计中很多常见分布指数分布、正态分布、分布、分布和分布的函数，利用函数转化会使计算简便有效；4、分布与标准正态分布之间的联系：验证了，随着样本容量的增加（），分布趋向于标准正态分布；一般认为，当时，分布接近于标准正态分布，此时按正态分布表来查就可以了，从而达到计算简化的效果，通过函数的性质、泰勒展开，用MATLAB对分布与标准正态分布的误差数值计算和模拟对比，从三种情况峰值比较、分布差异分析、密度函数值的比较验证了其结果的合理性。
 
 关键词：函数 分布 正态分布 MATLAB 泰勒展开

 Gamma function nature and application
 
Student: Yuting Advisor: Prof. Huyue
School of Science, Zhejiang University of Science and Technology
 
Abstract

 Gamma function in the classical mathematical, from the beginning of the systematic study Euler Gamma function has been more than 200 years, and beyond it as a function, with a rich and beautiful features, in many branches of mathematics and physics, engineering and other disciplines Have played an important role.
  In article main discussion: 1,Gamma function nature research; 2, explained through the example, the partial integration certain changes into Gamma the function from now on, as well as Gamma function and Beta function, trigonometric function relations; 3, in the probability statistics the common distribution index distribution, the normal distribution, the t distribution, the f distribution and  distribution's function, the use Gamma function transforms can cause the computation simple effective; 4, distributed and between standardized normal distribution relation: Has confirmed that along with sample size's increase（）, the t distribution tends to the standardized normal distribution; Generally, when, t distribution approaches in standard normal distribution, being able to according to checking t coming normal distribution form now , reached the simplified effect secretly scheming against thereby, the difference calculating and simulating contrast , secondary three kinds condition peak value to scatter comparatively, has analyzed , density function value comparison has verified whose result rationality by the fact that Gamma function character , Taylor spread out, with the error numerical value that MATLAB distributes to the t with standard normal distribution.

Keywords:  Gamma Function；t distribution；Normal distribution；MATLAB；Taylor launches

目  录 

摘  要 Ⅰ
Abstract  Ⅱ
目录  Ⅲ
引   言 1
1  问题的提出 2
2 基本概念和理论基础 3
2.1  基本概念 3
2.1.1  函数的概念 3
2.1.2  分布的基本概念 4
2.2  函数的性质 4
3  函数的应用 9
3.1  函数的其他形式和一个特殊值 9
3.1.1  函数常用的转化公式 9
3.1.2  函数的其他形式 10
3.1.2.1  函数和函数的关系 10
3.1.2.2  余元公式——函数与三角函数的关系 10
3.1.3  常见的概率分布 12
3.1.3.1  指数分布 12
3.1.3.2  分布 12
3.1.3.3  分布 13
3.1.3.4  分布 15
4  MATLAB工具的介绍 17
5  分布与标准正态分布之间的偏差 18
5.1  分布与标准正态分布之间的联系 18
5.1  分布与标准正态分布之间的偏差 19
6  结论 34
致 谢 35
参考文献 36
附件 37

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