A Generalized Wallis Formula

Abstract

This article generalizes the famous Wallis’s formula for k ≥ 0 , to an integral over the unit sphere Sn-1. An application to the integral of polynomials over Sn-1 is discussed.

Keywords

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Namazi, J. (2018) A Generalized Wallis Formula. Applied Mathematics, 9, 207-209. doi: 10.4236/am.2018.93015.

One of Wallis formulas is

${\int }_{0}^{\text{2π}}{\mathrm{sin}}^{2k}\theta \text{d}\theta ={\int }_{0}^{\text{2π}}{\mathrm{cos}}^{2k}\theta \text{d}\theta =\frac{\left(2k\right)!2\text{π}}{{2}^{2k}{\left(k!\right)}^{2}}$

for $k\ge 0$ . This formula can be proved by various methods [1] [2] [3] [4] including a repeated application of a reduction formula such as

${\int }_{0}^{\text{2π}}{\mathrm{sin}}^{k}\theta \text{d}\theta =\frac{k-1}{k}{\int }_{0}^{\text{2π}}{\mathrm{sin}}^{k-2}\theta \text{d}\theta$ . Note that $sin\theta$ and $cos\theta$ are coordinates

of a point on the unit sphere in R2. Since the above formula involves an integration over the unit circle in R2, its extension to higher dimensions is of interest.

For each $x=\left({x}_{1},{x}_{2},\cdots ,{x}_{n}\right)\in {R}^{n}$ , let $|x|={\left(\sum \text{ }{x}_{i}^{2}\right)}^{1/2}$ be its Euclidean norm. We call $\alpha =\left({\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{n}\right)$ , where ${\alpha }_{i}\ge 0$ are non-negative integers, a multi-index, and $|\alpha |=\sum |{\alpha }_{i}|$ its degree. Set $\alpha !={\alpha }_{1}!{\alpha }_{2}!\cdots {\alpha }_{n}!$ and ${x}^{\alpha }={x}_{1}^{{\alpha }_{1}}{x}_{2}^{{\alpha }_{2}}\cdots {x}_{n}^{{\alpha }_{n}}$ . Let $\text{ }{S}^{n-1}=\left\{\xi \in {R}^{n}:|\xi |=1\right\}$ be the unit sphere in Rn and $\text{d}\sigma$ be its surface measure. Let ${B}_{r}\left(a\right)=\left\{x\in {R}^{n}:|x-a|\le r\right\}$ stand for the ball of radius r centered at a. The gamma function is defined as $\Gamma \left(s\right)={\int }_{0}^{\infty }\text{ }{\text{e}}^{-t}{t}^{s-1}\text{d}t$ , for $s>0$ . The generalized Wallis’s formula is a special case of the following theorem.

Theorem 1 (i) ${\int }_{{S}^{n-1}}\text{ }{\xi }^{\alpha }\text{d}\sigma =0$ , if any ${\alpha }_{i}$ is odd. In particular, the integral equals zero if $|\alpha |$ is odd.

(ii) ${\int }_{{S}^{n-1}}\text{ }{\xi }^{2\alpha }\text{d}\sigma =\frac{\left(2\alpha \right)!2{\text{π}}^{n/2}}{{2}^{2|\alpha |}\alpha !\Gamma \left(n/2+|\alpha |\right)},\text{\hspace{0.17em}}|\alpha |\ge 0$ .

Setting ${\alpha }_{i}=k$ and ${\alpha }_{j}=0$ for $j\ne i$ in the theorem, the generalized Wallis’s formula follows

${\int }_{{S}^{n-1}}\text{ }\text{ }{\xi }_{i}^{2k}\text{d}\sigma =\frac{\left(2k\right)!2{\text{π}}^{n/2}}{{2}^{2k}k!\Gamma \left(n/2+k\right)},\text{\hspace{0.17em}}k\ge 0.$

Note that for $|\alpha |=0$ , (ii) is equivalent to the well-known formula

${\omega }_{n-1}=\frac{2{\text{π}}^{n/2}}{\Gamma \left(n/2\right)}$ (1)

where ${\omega }_{n-1}$ is the surface area of the unit sphere in Rn. Theorem 1 is interesting in its own right and has further applications. For example, for a polynomial

$p\left(x\right)={\sum }_{|\alpha |\le m}\text{ }{b}_{\alpha }{x}^{\alpha }$ of degree m, one may express ${\int }_{{B}_{r}\left(0\right)}\text{ }p\left(x\right)\text{d}x$ as a simple

polynomial of degree $n+m$ in r. In the following we use polar coordinates $x=\rho \xi ,\text{\hspace{0.17em}}\rho =|x|,\xi \in {S}^{n-1}$ .

$\begin{array}{c}{\int }_{{B}_{r}\left(0\right)}\text{ }\text{ }p\left(x\right)\text{d}x=\underset{|\alpha |\le m}{\sum }{b}_{\alpha }{\int }_{{B}_{r}\left(0\right)}\text{ }{x}^{\alpha }\text{d}x=\underset{|\alpha |\le m}{\sum }{b}_{\alpha }{\int }_{0}^{r}\text{ }{\rho }^{|\alpha |+n-1}\text{d}\rho {\int }_{{S}^{n-1}}\text{ }{\xi }^{\alpha }\text{d}\sigma \\ =\underset{2|\alpha |\le m}{\sum }\frac{{b}_{2\alpha }{r}^{2|\alpha |+n}}{2|\alpha |+n}{\int }_{{S}^{n-1}}\text{ }{\xi }^{2\alpha }\text{d}\sigma =\underset{|\alpha |\le \left[m/2\right]}{\sum }\frac{{b}_{2\alpha }{d}_{\alpha }}{2|\alpha |+n}{r}^{2k+n}\\ =\underset{k=0}{\overset{\left[m/2\right]}{\sum }}\left(\underset{|\alpha |=k}{\sum }\frac{{b}_{2\alpha }{d}_{\alpha }}{2k+n}\right){r}^{2k+n}=\underset{k=0}{\overset{\left[m/2\right]}{\sum }}\text{ }{c}_{k}{r}^{2k+n}.\end{array}$

Here ${d}_{\alpha }={\int }_{{S}^{n-1}}\text{ }{\xi }^{2\alpha }\text{d}\sigma$ as given by (ii), and [.] is the bracket function.

Proof of Theorem 1. (i) The proof is by induction on $|\alpha |$ .

If $|\alpha |=1$ then $\xi ={\xi }_{i}$ for some i. Therefore, ${\int }_{{S}^{n-1}}\text{ }{\xi }^{\alpha }\text{d}\sigma ={\int }_{{S}^{n-1}}\text{ }{\xi }_{i}\text{d}\sigma =0$ by the symmetry of the sphere.

Assume now the assertion is true for $|\alpha |\le m$ for some $m\ge 1$ . Let $|\alpha |=m+1$ and assume, without loss of generality, that ${\alpha }_{1}$ is odd. Applying the divergence theorem results in

$\begin{array}{c}{\int }_{{S}^{n-1}}\text{ }\text{ }{\xi }^{\alpha }\text{d}\sigma ={\int }_{{S}^{n-1}}\text{ }{\xi }_{1}\left({\xi }_{1}^{{\alpha }_{1}-1}{\xi }_{2}^{{\alpha }_{2}}\cdots {\xi }_{n}^{{\alpha }_{n}}\right)\text{d}\sigma \\ ={\int }_{{B}_{1}\left(0\right)}\frac{\partial }{\partial {x}_{1}}\left({x}_{1}^{{\alpha }_{1}-1}{x}_{2}^{{\alpha }_{2}}\cdots {x}_{n}^{{\alpha }_{n}}\right)\text{d}x.\end{array}$ (2)

If ${\alpha }_{1}=1$ , the last integral in (2) is zero. Otherwise, a conversion to polar coordinates in (2), yields,

$\begin{array}{c}{\int }_{{S}^{n-1}}\text{ }{\xi }^{\alpha }\text{d}\sigma =\left({\alpha }_{1}-1\right){\int }_{{B}_{1}\left(0\right)}\text{ }{x}_{1}^{{\alpha }_{1}-2}{x}_{2}^{{\alpha }_{2}}\cdots {x}_{n}^{{\alpha }_{n}}\text{d}x\\ =\left({\alpha }_{1}-1\right){\int }_{0}^{1}\text{ }{\rho }^{m+n-2}\text{d}\rho {\int }_{{S}^{n-1}}\text{ }{\xi }^{\beta }\text{d}\sigma =\frac{{\alpha }_{1}-1}{m+n-1}{\int }_{{S}^{n-1}}\text{ }{\xi }^{\beta }\text{d}\sigma ,\end{array}$

where $\beta =\left({\alpha }_{1}-2,{\alpha }_{2},\cdots ,{\alpha }_{n}\right)$ . The last integral is now zero, by the induction hypothesis.

ii) The proof is by induction on $|\alpha |$ .

For $|\alpha |=0$ , we must establish (1). Let ${e}_{n}={\int }_{{R}^{n}}\text{ }{\text{e}}^{-\text{π}{|x|}^{2}}\text{d}x\text{ }$ . Writing ${e}_{n}$ as a product of integrals and using polar coordinates in R2 followed by a change of variables, one obtains

$\begin{array}{c}{e}_{n}=\underset{i=1}{\overset{n}{\prod }}{\int }_{R}\text{ }{\text{e}}^{-\text{π}{x}_{i}^{2}}\text{d}{x}_{i}={\left({e}_{1}\right)}^{n}={\left({e}_{2}\right)}^{n/2}\\ ={\left({\int }_{0}^{\text{2π}}\text{ }\text{d}\theta {\int }_{0}^{\infty }r{\text{e}}^{-\text{π}{r}^{2}}\text{d}r\right)}^{n/2}={\left({\int }_{0}^{\infty }\text{ }{\text{e}}^{-u}\text{d}u\right)}^{n/2}=1.\end{array}$

We used a change of variable $u=\text{π}{r}^{2}$ in the previous integral. Converting to polar coordinates for Rn results in

$\begin{array}{c}1={e}_{n}={\int }_{{R}^{n}}\text{ }{\text{e}}^{-\text{π}{|x|}^{2}}\text{d}x={\omega }_{n-1}{\int }_{0}^{\infty }{r}^{n-1}{\text{e}}^{-\pi {r}^{2}}\text{d}r\\ =\frac{{\omega }_{n-1}}{2{\text{π}}^{n/2}}{\int }_{0}^{\infty }\text{ }{u}^{n/2-1}{\text{e}}^{-u}\text{d}u=\frac{\Gamma \left(n/2\right)}{2{\text{π}}^{n/2}}{\omega }_{n-1}.\end{array}$

Identity (1) follows immediately from the last equation.

Now suppose the claim is true for $|\alpha |=m$ . Let $|\alpha |=m+1$ . We may assume, without loss of generality, that ${\alpha }_{1}\ge 1$ . Applying the divergence theorem followed by a conversion to polar coordinates leads to

$\begin{array}{c}{\int }_{{S}^{n-1}}\text{ }{\xi }^{2\alpha }\text{d}\sigma ={\int }_{{S}^{n-1}}\text{ }{\xi }_{1}\left({\xi }_{1}^{2{\alpha }_{1}-1}{\xi }_{2}^{2{\alpha }_{2}}\cdots {\xi }_{n}^{2{\alpha }_{n}}\right)\text{d}\sigma ={\int }_{{B}_{1}\left(0\right)}\frac{\partial }{\partial {x}_{1}}\left({x}_{1}^{2{\alpha }_{1}-1}{x}_{2}^{2{\alpha }_{2}}\cdots {x}_{n}^{2{\alpha }_{n}}\right)\text{d}x\\ =\left(2{\alpha }_{1}-1\right){\int }_{{B}_{1}\left(0\right)}\text{ }{x}_{1}^{2{\alpha }_{1}-2}{x}_{2}^{2{\alpha }_{2}}\cdots {x}_{n}^{2{\alpha }_{n}}\text{d}x=\frac{2{\alpha }_{1}-1}{n+2m}{\int }_{{S}^{n-1}}\text{ }{\xi }^{2\beta }\text{d}\sigma ,\end{array}$

where $\beta =\left({\alpha }_{1}-1,{\alpha }_{2},\cdots ,{\alpha }_{n}\right)$ . Since $|\beta |=m$ , and using the fact that $\Gamma \left(s+1\right)=s\Gamma \left(s\right)$ along with the induction hypothesis, we get

${\int }_{{S}^{n-1}}\text{ }{\xi }^{2\alpha }\text{d}\sigma =\frac{2{\alpha }_{1}-1}{n+2m}.\frac{\left(2\beta \right)!2{\text{π}}^{n/2}}{{2}^{2|\beta |}\beta !\Gamma \left(n/2+m\right)}=\frac{\left(2\alpha \right)!2{\pi }^{n/2}}{{2}^{2|\alpha |}\alpha !\Gamma \left(n/2+|\alpha |\right)}.$

Conflicts of Interest

The authors declare no conflicts of interest.

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