In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and more precisely in
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
, the Wallis integrals constitute a family of
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s introduced by
John Wallis
John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...
.
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 (see below).
Recurrence relation
By means of
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
, a
reduction formula can be obtained. Using the identity
, we have for all
,
:
Integrating the second integral by parts, with:
:*
, whose
anti-derivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
is
:*
, whose
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
is
we have:
:
Substituting this result into equation (1) 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 integral
In mathematics, there are two types of Euler integral:
# The ''Euler integral of the first kind'' is the beta function \mathrm(z_1,z_2) = \int_0^1t^(1-t)^\,dt = \frac
# The ''Euler integral of the second kind'' is the gamma function \Gamma(z) = \i ...
s:
#''Euler
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
of the first kind'': the
Beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^(1 ...
:
#:
for
#''Euler integral of the second kind'': the
Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
:
#:
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
:
(equivalence of two sequences).
:Indeed, for all
:
:
(since the sequence is decreasing)
:
(since
)
:
(by equation
).
:By the
sandwich theorem
In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is trapped between two other functions.
The squeeze theorem is used in calculus and mathematical anal ...
, we conclude that
, and hence
.
*By examining
, one obtains the following equivalence:
:
(and consequently
).
Deducing Stirling's formula
Suppose that we have the following equivalence (known as
Stirling's formula
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less ...
):
:
for some constant
that we wish to determine. From above, we have
:
(equation (3))
Expanding
and using the formula above for the factorials, we get
:
From (3) and (4), 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
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is
\int_^\infty e^\,dx = \s ...
can be evaluated through the use of Wallis' integrals.
We first prove the following inequalities:
*
*
In fact, letting
,
the first inequality (in which