# Wallis formula

A formula which expresses the number $\pi /2$ as an infinite product:

$$\tag{1 } { \frac \pi {2} } = \ \left ( { \frac{2}{1} } \cdot { \frac{2}{3} } \right ) \left ( { \frac{4}{3} } \cdot { \frac{4}{5} } \right ) \dots \left ( { \frac{2k}{2k-1} } \cdot { \frac{2k}{2k+1} } \right ) \dots =$$

$$= \ \lim\limits _ {m \rightarrow \infty } \prod _ { k=1 } ^ { m } \frac{( 2k) ^ {2} }{( 2k- 1)( 2k+ 1) } .$$

There exist other variants of this formula, e.g.:

$$\tag{2 } \sqrt \pi = \ \lim\limits _ {m \rightarrow \infty } \ \frac{( m!) ^ {2} \cdot 2 ^ {2m} }{( 2m)! \sqrt m } .$$

Formula (1) was first employed by J. Wallis [1] in his calculation of the area of a disc; it is one of the earliest examples of an infinite product.

#### References

 [1] J. Wallis, "Arithmetica infinitorum" , Oxford (1656)

Formula (1) is a direct consequence of Euler's product formula

$$\sin z = z \prod _ { n=1 } ^ \infty \left ( 1 - \frac{z ^ {2} }{n ^ {2} \pi ^ {2} } \right )$$ with $z = \pi /2$.

It can also be obtained by expressing $\int _ {0} ^ {\pi /2 } \sin ^ {2m} x dx$ and $\int _ {0} ^ {\pi /2 } \sin ^ {2m+1} x dx$ in terms of $m$, and by showing that

$$\frac{\int\limits _ { 0 } ^ { \pi /2 } \sin ^ {2m} x dx }{\int\limits _ { 0 } ^ { \pi /2 } \sin ^ {2m+1} x dx } \rightarrow 1 \ ( m\rightarrow \infty ).$$

Formula (2) can be derived from (1) by multiplying the numerator and the denominator of $\prod _ {k=1} ^ {m} ( 2k) ^ {2} / ( 2k- 1)( 2k+ 1)$ by $2 ^ {2} \cdot 4 ^ {2} \dots ( 2m) ^ {2}$.

#### References

 [a1] T.M. Apostol, "Calculus" , 2 , Blaisdell (1964) [a2] C.H. Edwards jr., "The historical development of the calculus" , Springer (1979) [a3] P. Lax, S. Burstein, A. Lax, "Calculus with applications and computing" , 1 , Springer (1976) [a4] D.J. Struik (ed.) , A source book in mathematics: 1200–1800 , Harvard Univ. Press (1986)
How to Cite This Entry:
Wallis formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wallis_formula&oldid=51104
This article was adapted from an original article by T.Yu. Popova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article