应用Beta函数证明沃利斯公式的两种新证法

Two New Methods of Proving Wallis Formula by Using Beta Function

摘要: 沃利斯公式(Wallis Formula)是微积分中的一个重要的公式，是圆周率π的有理数极限表达式，其证明方法较多，一般都是利用积分∫₀^2/πsinⁿxdx=∫₀^2/πcosⁿxdx证明的。本文利用B函数以及B函数与Γ函数之间的关系，给出了沃利斯公式的两种新的证明方法。

Abstract: Wallis Formula is an important formula in calculus, which is the rational limit expression of PI. There are many methods to prove it, which are generally proved by integral ∫₀^2/πsinⁿxdx=∫₀^2/πcosⁿxdx. In this paper, two new methods of proving Wallis Formula are given by using Beta Function and the relationship between Beta Function and Gamma Function.