Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

An Improper Integral with a Square Root

Version 1 : Received: 20 October 2016 / Approved: 21 October 2016 / Online: 21 October 2016 (08:29:08 CEST)

A peer-reviewed article of this Preprint also exists.

Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza--Issues of Analysis 7 (25) (2018), no. 1, 104--115; Available online at https://doi.org/10.15393/j3.art.2018.4370. Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza--Issues of Analysis 7 (25) (2018), no. 1, 104--115; Available online at https://doi.org/10.15393/j3.art.2018.4370.

Journal reference: Problemy Analiza--Issues of Analysis 2018, 7, 104-115
DOI: 10.15393/j3.art.2018.4370

Abstract

In the paper, the author presents explicit and unified expressions for a sequence of improper integrals in terms of the beta functions and the Wallis ratios. Hereafter, the author derives integral representations for the Catalan numbers originating from combinatorics.

Keywords

improper integral; explicit expression; unified expression; beta function; Wallis ratio; integral representation; Catalan number

Subject

MATHEMATICS & COMPUTER SCIENCE, Analysis

Comments (6)

Comment 1
Received: 22 April 2018
Commenter: Feng Qi (Click to see Publons profile: )
The commenter has declared there is no conflict of interests.
Comment: This preprint has been accepted for publication as

Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza--Issues of Analysis 7 (25) (2018), in press; Available online at https://doi.org/10.15393/j3.art.2018.????
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Comment 2
Received: 11 June 2018
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the author
Comment: This preprint has been accepted for publication as

Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza--Issues of Analysis 7 (25) (2018), no. 1, in press; Available online at https://doi.org/10.15393/j3.art.2018.4370
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Comment 3
Received: 28 June 2018
Commenter: Feng Qi (Click to see Publons profile: )
Commenter's Conflict of Interests: I am the author
Comment: Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy AnalizaIssues of Analysis 7 (25) (2018), no. 1, 104115; Available online at https://doi.org/10.15393/j3.art.2018.4370
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Comment 4
Received: 28 June 2018
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the author
Comment: Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza—Issues of Analysis 7 (25) (2018), no. 1, 104115; Available online at https://doi.org/10.15393/j3.art.2018.4370
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Comment 5
Received: 28 June 2018
Commenter: Feng Qi (Click to see Publons profile: )
Commenter's Conflict of Interests: I am the author
Comment: This preprint has been formally published as

Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza—Issues of Analysis 7 (25) (2018), no. 1, 104—115; Available online at https://doi.org/10.15393/j3.art.2018.4370
+ Respond to this comment
Comment 6
Received: 28 June 2018
Commenter: Feng Qi (Click to see Publons profile: )
Commenter's Conflict of Interests: I am the author
Comment: This preprint has been formally published as

Feng Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza—Issues of Analysis 7 (25) (2018), no. 1, 104—115; Available online at https://doi.org/10.15393/j3.art.2018.4370
+ Respond to this comment

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