Wallis formula

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A formula which expresses the number as an infinite product:


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


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.


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


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

with .

It can also be obtained by expressing and in terms of , and by showing that

Formula (2) can be derived from (1) by multiplying the numerator and the denominator of by .


[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. T.Yu. Popova (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098