Javad Namazi
Fairleigh Dickinson University, Madison, NJ, USA.
DOI: 10.4236/am.2018.93015   PDF    HTML   XML   1,177 Downloads   5,194 Views   Citations

Abstract

Keywords

Willis Formula, Unit Sphere

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One of Wallis formulas is

${\displaystyle {\int }_0^{\text{2π}}}{\sin }^{2k}\theta \text{d}\theta ={\displaystyle {\int }_0^{\text{2π}}}{\cos }^{2k}\theta \text{d}\theta =\frac{\left(2k\right)!2\text{π}}{2^{2k}{\left(k!\right)}^2}$

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

${\displaystyle {\int }_0^{\text{2π}}}{\sin }^k\theta \text{d}\theta =\frac{k-1}{k}{\displaystyle {\int }_0^{\text{2π}}}{\sin }^{k-2}\theta \text{d}\theta$ . 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 $x=\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_n\right)\in R^n$ , let $\left|x\right|={\left({\displaystyle \sum }{x}_i^2\right)}^{1/2}$ 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 $\left|\alpha \right|={\displaystyle \sum }\left|{\alpha }_i\right|$ 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 $S^{n-1}=\left\{\xi \in R^n:\left|\xi \right|=1\right\}$ be the unit sphere in Rⁿ and $\text{d}\sigma$ be its surface measure. Let $B_r\left(a\right)=\left\{x\in R^n:\left|x-a\right|\le r\right\}$ stand for the ball of radius r centered at a. The gamma function is defined as $\Gamma \left(s\right)={\displaystyle {\int }_0^{\infty }}{\text{e}}^{-t}{t}^{s-1}\text{d}t$ , for $s>0$ . The generalized Wallis’s formula is a special case of the following theorem.

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

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

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

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

Note that for $\left|\alpha \right|=0$ , (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

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

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

$\begin{array}{c}{\displaystyle {\int }_{B_r\left(0\right)}}p\left(x\right)\text{d}x={\displaystyle \underset{\left|\alpha \right|\le m}{\sum }}{b}_{\alpha }{\displaystyle {\int }_{B_r\left(0\right)}}{x}^{\alpha }\text{d}x={\displaystyle \underset{\left|\alpha \right|\le m}{\sum }}{b}_{\alpha }{\displaystyle {\int }_0^r}{\rho }^{\left|\alpha \right|+n-1}\text{d}\rho {\displaystyle {\int }_{S^{n-1}}}{\xi }^{\alpha }\text{d}\sigma \\ ={\displaystyle \underset{2\left|\alpha \right|\le m}{\sum }}\frac{b_{2\alpha }{r}^{2\left|\alpha \right|+n}}{2\left|\alpha \right|+n}{\displaystyle {\int }_{S^{n-1}}}{\xi }^{2\alpha }\text{d}\sigma ={\displaystyle \underset{\left|\alpha \right|\le \left[m/2\right]}{\sum }}\frac{b_{2\alpha }{d}_{\alpha }}{2\left|\alpha \right|+n}{r}^{2k+n}\\ ={\displaystyle \underset{k=0}{\overset{\left[m/2\right]}{\sum }}}\left({\displaystyle \underset{\left|\alpha \right|=k}{\sum }}\frac{b_{2\alpha }{d}_{\alpha }}{2k+n}\right)r^{2k+n}={\displaystyle \underset{k=0}{\overset{\left[m/2\right]}{\sum }}}{c}_k{r}^{2k+n}.\end{array}$

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

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

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

Assume now the assertion is true for $\left|\alpha \right|\le m$ for some $m\ge 1$ . Let $\left|\alpha \right|=m+1$ and assume, without loss of generality, that ${\alpha }_1$ is odd. Applying the divergence theorem results in

$\begin{array}{c}{\displaystyle {\int }_{S^{n-1}}}{\xi }^{\alpha }\text{d}\sigma ={\displaystyle {\int }_{S^{n-1}}}{\xi }_1\left({\xi }_1^{{\alpha }_1-1}{\xi }_2^{{\alpha }_2}\cdots {\xi }_n^{{\alpha }_n}\right)\text{d}\sigma \\ ={\displaystyle {\int }_{B_1\left(0\right)}}\frac{\partial }{\partial x_1}\left(x_1^{{\alpha }_1-1}{x}_2^{{\alpha }_2}\cdots x_n^{{\alpha }_n}\right)\text{d}x.\end{array}$ (2)

If ${\alpha }_1=1$ , the last integral in (2) is zero. Otherwise, a conversion to polar coordinates in (2), yields,

$\begin{array}{c}{\displaystyle {\int }_{S^{n-1}}}{\xi }^{\alpha }\text{d}\sigma =\left({\alpha }_1-1\right){\displaystyle {\int }_{B_1\left(0\right)}}{x}_1^{{\alpha }_1-2}{x}_2^{{\alpha }_2}\cdots x_n^{{\alpha }_n}\text{d}x\\ =\left({\alpha }_1-1\right){\displaystyle {\int }_0^1}{\rho }^{m+n-2}\text{d}\rho {\displaystyle {\int }_{S^{n-1}}}{\xi }^{\beta }\text{d}\sigma =\frac{{\alpha }_1-1}{m+n-1}{\displaystyle {\int }_{S^{n-1}}}{\xi }^{\beta }\text{d}\sigma ,\end{array}$

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 $\left|\alpha \right|$ .

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

$\begin{array}{c}{e}_n={\displaystyle \underset{i=1}{\overset{n}{\prod }}}{\displaystyle {\int }_R}{\text{e}}^{-\text{π}{x}_i^2}\text{d}{x}_i={\left(e_1\right)}^n={\left(e_2\right)}^{n/2}\\ ={\left({\displaystyle {\int }_0^{\text{2π}}}\text{d}\theta {\displaystyle {\int }_0^{\infty }}r{\text{e}}^{-\text{π}{r}^2}\text{d}r\right)}^{n/2}={\left({\displaystyle {\int }_0^{\infty }}{\text{e}}^{-u}\text{d}u\right)}^{n/2}=1.\end{array}$

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

$\begin{array}{c}1=e_n={\displaystyle {\int }_{R^n}}{\text{e}}^{-\text{π}{\left|x\right|}^2}\text{d}x={\omega }_{n-1}{\displaystyle {\int }_0^{\infty }}{r}^{n-1}{\text{e}}^{-\pi r^2}\text{d}r\\ =\frac{{\omega }_{n-1}}{2{\text{π}}^{n/2}}{\displaystyle {\int }_0^{\infty }}{u}^{n/2-1}{\text{e}}^{-u}\text{d}u=\frac{\Gamma \left(n/2\right)}{2{\text{π}}^{n/2}}{\omega }_{n-1}.\end{array}$

Identity (1) follows immediately from the last equation.

Now suppose the claim is true for $\left|\alpha \right|=m$ . Let $\left|\alpha \right|=m+1$ . 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

$\begin{array}{c}{\displaystyle {\int }_{S^{n-1}}}{\xi }^{2\alpha }\text{d}\sigma ={\displaystyle {\int }_{S^{n-1}}}{\xi }_1\left({\xi }_1^{2{\alpha }_1-1}{\xi }_2^{2{\alpha }_2}\cdots {\xi }_n^{2{\alpha }_n}\right)\text{d}\sigma ={\displaystyle {\int }_{B_1\left(0\right)}}\frac{\partial }{\partial x_1}\left(x_1^{2{\alpha }_1-1}{x}_2^{2{\alpha }_2}\cdots x_n^{2{\alpha }_n}\right)\text{d}x\\ =\left(2{\alpha }_1-1\right){\displaystyle {\int }_{B_1\left(0\right)}}{x}_1^{2{\alpha }_1-2}{x}_2^{2{\alpha }_2}\cdots x_n^{2{\alpha }_n}\text{d}x=\frac{2{\alpha }_1-1}{n+2m}{\displaystyle {\int }_{S^{n-1}}}{\xi }^{2\beta }\text{d}\sigma ,\end{array}$

where $\beta =\left({\alpha }_1-1,{\alpha }_2,\cdots ,{\alpha }_n\right)$ . Since $\left|\beta \right|=m$ , and using the fact that $\Gamma \left(s+1\right)=s\Gamma \left(s\right)$ along with the induction hypothesis, we get

${\displaystyle {\int }_{S^{n-1}}}{\xi }^{2\alpha }\text{d}\sigma =\frac{2{\alpha }_1-1}{n+2m}.\frac{\left(2\beta \right)!2{\text{π}}^{n/2}}{2^{2\left|\beta \right|}\beta !\Gamma \left(n/2+m\right)}=\frac{\left(2\alpha \right)!2{\pi }^{n/2}}{2^{2\left|\alpha \right|}\alpha !\Gamma \left(n/2+\left|\alpha \right|\right)}.$

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Muller, C. (1966) Spherical Harmonics. Springer-Verlag Berlin Heidelberg. [CrossRef]
[2]	Macrobert, T.M. (1948) Spherical Harmonics: An Elementary Treatise on Harmonic Functions with Applications. Dover Publications, New York.
[3]	Stein, E. and Weiss, G. (1971) Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton.
[4]	Jeffrey, H. and Jeffrey, B. (1999) Methods of Mathematical Physics. 3rd Edition, Cambridge University Press, Cambridge.