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A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex FunctionsAsymptotic Analysis

A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions: Asymptotic Analysis [The asymptotic behavior of the gamma function for large values of its argument can be summarized as follows: for any a ≥ 0, we have the following asymptotic equivalences (see Titchmarsh (The Theory of Functions, 2nd edn. Oxford University Press, Oxford, 1939, Section 1.87)) Γ(x+a)∼xaΓ(x)asx→∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \begin{aligned} \Gamma (x+a) ~\sim ~ x^a\,\Gamma (x) \mbox{as }x\to \infty {\,},\end{aligned} $$ \end{document}Γ(x)∼2πe−xxx−12asx→∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \begin{aligned}\Gamma (x) ~\sim ~ \sqrt {2\pi }{\,}e^{-x}x^{x-\frac {1}{2}} \mbox{as }x\to \infty {\,},\end{aligned} $$ \end{document}Γ(x+1)∼2πxe−xxxasx→∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \begin{aligned}\Gamma (x+1) ~\sim ~ \sqrt {2\pi x}{\,}e^{-x}x^x \mbox{as }x\to \infty {\,}, \end{aligned} $$ \end{document} where both formulas (6.2) and (6.3) are known by the name Stirling’s formula.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex FunctionsAsymptotic Analysis

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References (1)

  • T M Apostol (1999)

    409

    Amer Math Monthly, 106

Publisher
Springer International Publishing
Copyright
© The Editor(s) (if applicable) and The Author(s) 2022. This book is an open access publication.
ISBN
978-3-030-95087-3
Pages
59 –89
DOI
10.1007/978-3-030-95088-0_6
Publisher site
See Chapter on Publisher Site

Abstract

[The asymptotic behavior of the gamma function for large values of its argument can be summarized as follows: for any a ≥ 0, we have the following asymptotic equivalences (see Titchmarsh (The Theory of Functions, 2nd edn. Oxford University Press, Oxford, 1939, Section 1.87)) Γ(x+a)∼xaΓ(x)asx→∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \begin{aligned} \Gamma (x+a) ~\sim ~ x^a\,\Gamma (x) \mbox{as }x\to \infty {\,},\end{aligned} $$ \end{document}Γ(x)∼2πe−xxx−12asx→∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \begin{aligned}\Gamma (x) ~\sim ~ \sqrt {2\pi }{\,}e^{-x}x^{x-\frac {1}{2}} \mbox{as }x\to \infty {\,},\end{aligned} $$ \end{document}Γ(x+1)∼2πxe−xxxasx→∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \begin{aligned}\Gamma (x+1) ~\sim ~ \sqrt {2\pi x}{\,}e^{-x}x^x \mbox{as }x\to \infty {\,}, \end{aligned} $$ \end{document} where both formulas (6.2) and (6.3) are known by the name Stirling’s formula.]

Published: Mar 10, 2022

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