Definition, basic properties
The first few terms of this sequence are:
Integrating the second integral by parts, with:
Substituting this result into equation (1) gives
Another relation to evaluate the Wallis' integrals
Wallis's integrals can be evaluated by using Euler integrals:
so this gives us the following relation to evaluate the Wallis integrals:
Deducing Stirling's formula
Suppose that we have the following equivalence (known as Stirling's formula):
From (3) and (4), we obtain by transitivity:
Evaluating the Gaussian Integral
The Gaussian integral can be evaluated through the use of Wallis' integrals.
We first prove the following inequalities:
Remark: There are other methods of evaluating the Gaussian integral.
Some of them are more direct.
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Wallis' integrals.