Special Functions: The Gamma and Beta Functions
Special Functions: The Gamma and Beta Functions
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特殊函数：Gamma 和 Beta 函数
DOI：
10.1017/cbo9781107325937.002
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发表时间：
1978
影响因子：
1.1
通讯作者：
R. Roy
中科院分区：
文献类型：
--
作者：
G. Andrews;R. Askey;R. Roy
欧拉在扩展阶乘函数的定义域时发现了伽玛函数，即Γ（x）。因此，Γ（x）是等于（x − 1）的亚纯函数！当x是正整数时。伽玛函数有几种表示法，但最重要的两种，由欧拉发现，将其表示为无限积分和有限乘积的极限。我们把第二个定义。与其将贝塔函数视为函数，不如将其视为一类积分--可以用伽马函数计算的积分--更有启发性。因此，我们经常把beta函数称为beta积分。在这一章中，我们发展了beta和gamma函数的一些基本性质。对某些结果给出了多个证明。通常，一个证明可以推广，而其他证明则不能。我们简要地讨论了有限域类似的伽玛和β函数。这些被称为高斯和雅可比和，在数论中很重要。我们展示了如何可以用它们来证明费马定理，一个素数的形式4 n + 1是可表示为两个平方和。我们还治疗一个简单的多维扩展的β积分，由于狄利克雷，从其中的体积的n维椭球体可以推导出来。
Euler discovered the gamma function, Γ( x ), when he extended the domain of the factorial function. Thus Γ( x ) is a meromorphic function equal to ( x − 1)! when x is a positive integer. The gamma function has several representations, but the two most important, found by Euler, represent it as an infinite integral and as a limit of a finite product. We take the second as the definition. Instead of viewing the beta function as a function, it is more illuminating to think of it as a class of integrals – integrals that can be evaluated in terms of gamma functions. We therefore often refer to beta functions as beta integrals. In this chapter, we develop some elementary properties of the beta and gamma functions. We give more than one proof for some results. Often, one proof generalizes and others do not. We briefly discuss the finite field analogs of the gamma and beta functions. These are called Gauss and Jacobi sums and are important in number theory. We show how they can be used to prove Fermat's theorem that a prime of the form 4 n + 1 is expressible as a sum of two squares. We also treat a simple multidimensional extension of a beta integral, due to Dirichlet, from which the volume of an n -dimensional ellipsoid can be deduced.