월리스 공식

월리스 공식()은 원주율을 구하는 간단한 공식으로, 존 월리스에 의해 만들어졌다.

Proof using integration 
Wallis derived this infinite product as it is done in calculus books today, 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:

We obtain values for  and  for later use.

Now, we calculate for even values  by repeatedly applying the recurrence relation result from the integration by parts.  Eventually, we end get down to , which we have calculated.

Repeating the process for odd values ,

We make the following observation, based on the fact that 

Dividing by :

, where the equality comes from our recurrence relation.

By the squeeze theorem,

같이 보기 
 비에트의 공식
 라이프니츠의 공식
 아르키메데스의 공식

원주율 알고리즘