# Wallis' integrals

In mathematics, and more precisely in analysis, the Wallis integrals constitute a family of integrals introduced by John Wallis. Wallis' integrals_sentence_0

## Definition, basic properties Wallis' integrals_section_0

The first few terms of this sequence are: Wallis' integrals_sentence_1

## Recurrence relation Wallis' integrals_section_1

Integrating the second integral by parts, with: Wallis' integrals_sentence_2

we have: Wallis' integrals_sentence_3

Substituting this result into equation (1) gives Wallis' integrals_sentence_4

and thus Wallis' integrals_sentence_5

## Another relation to evaluate the Wallis' integrals Wallis' integrals_section_2

Wallis's integrals can be evaluated by using Euler integrals: Wallis' integrals_sentence_6

so this gives us the following relation to evaluate the Wallis integrals: Wallis' integrals_sentence_7

## Deducing Stirling's formula Wallis' integrals_section_4

Suppose that we have the following equivalence (known as Stirling's formula): Wallis' integrals_sentence_8

From (3) and (4), we obtain by transitivity: Wallis' integrals_sentence_9

## Evaluating the Gaussian Integral Wallis' integrals_section_5

The Gaussian integral can be evaluated through the use of Wallis' integrals. Wallis' integrals_sentence_10

We first prove the following inequalities: Wallis' integrals_sentence_11

Remark: There are other methods of evaluating the Gaussian integral. Wallis' integrals_sentence_12

Some of them are more direct. Wallis' integrals_sentence_13