Wallis product

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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)

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

Proof using integration[edit]

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 method[edit]

See the main page on Gaussian integral.

Proof using Euler's infinite product for the sine function[edit]

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 approximation[edit]

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 zero[edit]

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 also[edit]


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

External links[edit]