Applied Mathematics
Vol.09 No.03(2018), Article ID:83199,3 pages
10.4236/am.2018.93015
A Generalized Wallis Formula
Javad Namazi
Fairleigh Dickinson University, Madison, NJ, USA
Copyright © 2018 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: February 7, 2018; Accepted: March 19, 2018; Published: March 22, 2018
ABSTRACT
This article generalizes the famous Wallis’s formula for , to an integral over the unit sphere . An application to the integral of polynomials over is discussed.
Keywords:
Willis Formula, Unit Sphere
One of Wallis formulas is
for . This formula can be proved by various methods [1] [2] [3] [4] including a repeated application of a reduction formula such as
. Note that and 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 , let be its Euclidean norm. We call , where are non-negative integers, a multi-index, and its degree. Set and . Let be the unit sphere in Rn and be its surface measure. Let stand for the ball of radius r centered at a. The gamma function is defined as , for . The generalized Wallis’s formula is a special case of the following theorem.
Theorem 1 (i) , if any is odd. In particular, the integral equals zero if is odd.
(ii) .
Setting and for in the theorem, the generalized Wallis’s formula follows
Note that for , (ii) is equivalent to the well-known formula
(1)
where 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
of degree m, one may express as a simple
polynomial of degree in r. In the following we use polar coordinates .
Here as given by (ii), and [.] is the bracket function.
Proof of Theorem 1. (i) The proof is by induction on .
If then for some i. Therefore, by the symmetry of the sphere.
Assume now the assertion is true for for some . Let and assume, without loss of generality, that is odd. Applying the divergence theorem results in
(2)
If , the last integral in (2) is zero. Otherwise, a conversion to polar coordinates in (2), yields,
where . The last integral is now zero, by the induction hypothesis.
ii) The proof is by induction on .
For , we must establish (1). Let . Writing as a product of integrals and using polar coordinates in R2 followed by a change of variables, one obtains
We used a change of variable in the previous integral. Converting to polar coordinates for Rn results in
Identity (1) follows immediately from the last equation.
Now suppose the claim is true for . Let . We may assume, without loss of generality, that . Applying the divergence theorem followed by a conversion to polar coordinates leads to
where . Since , and using the fact that along with the induction hypothesis, we get
Cite this paper
Namazi, J. (2018) A Generalized Wallis Formula. Applied Mathematics, 9, 207-209. https://doi.org/10.4236/am.2018.93015
References
- 1. Muller, C. (1966) Spherical Harmonics. Springer-Verlag Berlin Heidelberg. https://doi.org/10.1007/BFb0094775
- 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.