Wallis' integrals

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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

Equivalence Wallis' integrals_section_3

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

Note Wallis' integrals_section_6

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Wallis' integrals.