In mathematics, the Wallis product for , published in 1656 by John Wallis, states that
Overview
In mathematics, the Wallis product for , published in 1656 by John Wallis, states that
Proof using integration
Wallis derived this infinite product using interpolation, though his method is not regarded as rigorous. A modern derivation can be found by examining for even and odd values of , and noting that for large , increasing by 1 results in a change that becomes ever smaller as increases. Let
(This is a form of Wallis' integrals.) Integrate by parts:
Now, we make two variable substitutions for convenience to obtain:
