Wallis' integrals

In mathematics, and more precisely in analysis, the Wallis integrals constitute a family of integrals introduced by John Wallis.

Definition, basic properties

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. In fact, for all

because it is an integral of a non-negative continuous function which is not identically zero;
again because the last integral is of a non-negative continuous function.

Since the sequence is decreasing and bounded below by 0, it converges to a non-negative limit. Indeed, the limit is zero.

Recurrence relation

By means of integration by parts, a reduction formula can be obtained. Using the identity, we have for all,
Integrating the second integral by parts, with:
we have:
Substituting this result into equation gives
and thus
for all
This is a recurrence relation giving in terms of. This, together with the values of and give us two sets of formulae for the terms in the sequence, depending on whether is odd or even:
*

Another relation to evaluate the Wallis' integrals

Wallis's integrals can be evaluated by using Euler integrals:

Euler integral of the first kind: the Beta function:
: for
Euler integral of the second kind: the Gamma function:
: for.

If we make the following substitution inside the Beta function:
we obtain:
so this gives us the following relation to evaluate the Wallis integrals:
So, for odd, writing, we have:
whereas for even, writing and knowing that, we get :

Equivalence

From the recurrence formula above, we can deduce that
By examining, one obtains the following equivalence:

Deducing Stirling's formula

Suppose that we have the following equivalence :
for some constant that we wish to determine. From above, we have
Expanding and using the formula above for the factorials, we get
From and, we obtain by transitivity:
Solving for gives In other words,

Deducing the Double Factorial Ratio

Similarly, from above, we have:
Expanding and using the formula above for double factorials, we get:
Simplifying, we obtain:
or

Evaluating the Gaussian Integral

The Gaussian integral can be evaluated through the use of Wallis' integrals.
We first prove the following inequalities:
In fact, letting,
the first inequality is
equivalent to ;
whereas the second inequality reduces to
which becomes.
These 2 latter inequalities follow from the convexity of the
exponential function
.
Letting and
making use of the basic properties of improper integrals
,
we obtain the inequalities:
for use with the sandwich theorem.
The first and last integrals can be evaluated easily using
Wallis' integrals.
For the first one, let
.
Then, the integral becomes.
For the last integral, let
.
Then, it becomes.
As we have shown before,
. So, it follows that
Remark: There are other methods of evaluating the Gaussian integral.
Some of them are more direct.

Note

The same properties lead to Wallis product,
which expresses
in the form of an infinite product.