In mathematics, and more precisely in analysis, the Wallis integrals constitute a family of integrals introduced by John Wallis.
The Wallis integrals are the terms of the sequence
defined by or equivalently, The first few terms of this sequence are: The sequence
is decreasing and has positive terms.
is decreasing and bounded below by 0, it converges to a non-negative limit.
By means of integration by parts, a reduction formula can be obtained.
, Integrating the second integral by parts, with: we have: Substituting this result into equation (1) gives and thus for all
This is a recurrence relation giving
give us two sets of formulae for the terms in the sequence
is odd or even: Wallis's integrals can be evaluated by using Euler integrals: If we make the following substitution inside the Beta function:
sin
d t = 2 sin u cos u d u
we obtain: so this gives us the following relation to evaluate the Wallis integrals: So, for odd
, we have, by the product rules of equivalents,
, from which the desired result follows (noting that
Suppose that we have the following equivalence (known as Stirling's formula): for some constant
and using the formula above for the factorials, we get From (3) and (4), we obtain by transitivity: Solving for
In other words, Similarly, from above, we have: Expanding
and using the formula above for double factorials, we get: Simplifying, we obtain: or The Gaussian integral can be evaluated through the use of Wallis' integrals.
We first prove the following inequalities: In fact, letting
; whereas the second inequality reduces to
These 2 latter inequalities follow from the convexity of the exponential function (or from an analysis of the function
and making use of the basic properties of improper integrals (the convergence of the integrals is obvious), we obtain the inequalities:
for use with the sandwich theorem (as
The first and last integrals can be evaluated easily using Wallis' integrals.
sin
For the last integral, let
Remark: There are other methods of evaluating the Gaussian integral.
The same properties lead to Wallis product, which expresses
) in the form of an infinite product.