# Wallis formula

From Encyclopedia of Mathematics

A formula which expresses the number as an infinite product:

(1) |

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

(2) |

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) |

#### Comments

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 .

#### 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. T.Yu. Popova (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Wallis_formula&oldid=13195

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098