Wallis product


In mathematics, the Wallis product for π, published in 1656 by John Wallis,[1] states that

Comparison of the convergence of the Wallis product (purple asterisks) and several historical infinite series for π. Sn is the approximation after taking n terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times. (click for detail)

Proof using integrationEdit

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[2]


(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,


Proof using Laplace's methodEdit

See the main page on Gaussian integral.

Proof using Euler's infinite product for the sine functionEdit

While the proof above is typically featured in modern calculus textbooks, the Wallis product is, in retrospect, an easy corollary of the later Euler infinite product for the sine function.


Let  :


Relation to Stirling's approximationEdit

Stirling's approximation for the factorial function   asserts that


Consider now the finite approximations to the Wallis product, obtained by taking the first   terms in the product


where   can be written as


Substituting Stirling's approximation in this expression (both for   and  ) one can deduce (after a short calculation) that   converges to   as  .

Derivative of the Riemann zeta function at zeroEdit

The Riemann zeta function and the Dirichlet eta function can be defined:[1]


Applying an Euler transform to the latter series, the following is obtained:


See alsoEdit


  1. ^ a b c "Wallis Formula".
  2. ^ "Integrating Powers and Product of Sines and Cosines: Challenging Problems".

External linksEdit