Wallis product

The Wallis product is the infinite product representation of π: It was published in 1656 by John Wallis.

[1] Wallis derived this infinite product using interpolation, though his method is not regarded as rigorous.

A modern derivation can be found by examining

by 1 results in a change that becomes ever smaller as

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

Now, we calculate for even values

by repeatedly applying the recurrence relation result from the integration by parts.

Repeating the process for odd values

, We make the following observation, based on the fact that

: By the squeeze theorem, See the main page on Gaussian integral.

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.

: Stirling's approximation for the factorial function

asserts that Consider now the finite approximations to the Wallis product, obtained by taking the first

can be written as Substituting Stirling's approximation in this expression (both for

) one can deduce (after a short calculation) that

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: