Preprint Article Version 1 This version is not peer-reviewed

An Improper Integral with a Square Root

Feng Qi * ORCID logo
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.

Subject Areas

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

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Readers' Comments and Ratings (6)

Comment 1
Received: 22 April 2018
Commenter: Feng Qi
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
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
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
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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