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About 10,000 papers for “Gamma function”
Xiao-ming Zhang, F. Qi, Tie-quan Xu et al. · 2006
In this paper, the geometric convexity of a function involving gamma function is studied, as applications to inequality theory, some important inequalities which improve some known inequalities, including Wallis ’ inequality, are obtained.
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I. G. Zaltsman · A-to-Z Guide to Thermodynamics Heat and Mass Transfer and Fluids Engineering · 2008
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Lukong Cornelius Fai · 2024
This chapter examines the so-called Eulerian functions, namely the well-known Gamma function and Beta function, followed by some special functions related to them. This chapter also introduces a series of functional forms attributed to the Gamma or Euler function (discovered by the Swiss mathematician Leonhard Euler in 1729 in an attempt to find an analytical continuation of the factorial function), denoted by Γ(<i>z</i>) (notation introduced by Adrien–Marie Legendre in 1814) with argument <i>z</i> where analytical tools are employed consistent with the multiplicative Fourier transform or the so-called Mellin transform.
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Mark Ladd · 1998
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George B. Arfken · Elsevier eBooks · 1985
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David Ullrich · Graduate studies in mathematics · 2008
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Wolfram Koepf · Universitext · 2014
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2013
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Probability and Mathematical Statistics · 1972
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Bill Casselman · 2014
I attmept here a somewhat unorthodox introduction to the Gamma function. My principal references here are [Schwartz:1965]. and [Tate:1950/1967]. Contents 1. Characters as distributions 2. Parties finies 3. The Gamma function 4. The volumes and areas of spheres
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Avram Sidi · 2003
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Ranjan Roy · Cambridge University Press eBooks · 2021
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Charles H. C. Little, Kee L. Teo, Bruce van Brunt · Springer undergraduate mathematics series · 2021
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Cedric A. B. Smith · The Mathematical Gazette · 1939
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Benedict Irwin · 2026
Consider the integral $$ G(n) = \int_0^\infty\int_0^\infty \int_0^\infty e^{-x_1-x_2-x_3}(|x_1-x_2|,|x_2-x_3|,|x_3-x_1|)^n\; dx_1 dx_2 dx_3 $$ this is somewhat analagous to the gamma function which could be written $$ \Gamma(n+1) = \int_0^\infty \int_0^\infty e^{-x_1-x_2}(|x_1-x_2|)^n\; dx_1 dx_2 $$ nice values which seem to be related to Mersenne products n G(n) Mersenne --- --------- ------------------------------------- 0 1 (2¹ − 1) 1 3/2 (2² − 1)/2 2 7/2 (2³ − 1)/2 3 45/4 (2⁴ − 1)(2² − 1)/4 4 93/2 (2⁵ − 1)(2² − 1)/2 5 945/4 (2⁶ − 1)(2⁴ − 1)/4 6 5715/4 (2⁷ − 1)(2⁴ − 1)(2² − 1)/4 7 80325/8 (2⁸ − 1)(2⁶ − 1)(2⁴ − 1)/(2² − 1)/8 the denominators appear to relate to A001316 called Gould’s sequence, which is given by $$ d_n = ^n \left[{k} \bmod 2\right] $$ the numerators do not appear to be in the OEIS, but appear to be divisible by 2n + 1 − 1. The part left seems to be the largest odd divisor of n!, or related. From this it seems that $$ G(n) = \int_0^\infty\int_0^\infty \int_0^\infty e^{-x_1-x_2-x_3}(|x_1-x_2|,|x_2-x_3|,|x_3-x_1|)^n\; dx_1 dx_2 dx_3 = -1)n!}{2^n} = n!(2-2^{-n}) $$ which also seems to generalise to $$ G(n) = \int_0^\infty\int_0^\infty \int_0^\infty e^{-x_1-x_2-x_3}(|x_1-x_2|,|x_2-x_3|,|x_3-x_1|)^n\; dx_1 dx_2 dx_3 = \Gamma(n+1)(2-2^{-n}) $$ for non integer n. For drawing k variables from the distribution, and taking the max of all ${2}$ variables, we appear to have $$ G_k(n) = (k-1) \int_0^\infty x^{n} e^{-(k-1)x}(e^{x}-1)^{k-2} \; dx $$ which seems to be asymptotic to log(k). In fact Gk(1)−log(k) seems to approach γ
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Henryk Iwaniec · University lecture series · 2014
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Nikolai Sinitsyn, Valery Pokrovsky · 2026
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Artin, Emil, Butler, Michael · CERN Bulletin · 2015
This brief monograph on the gamma function was designed by the author to fill what he perceived as a gap in the literature of mathematics, which often treated the gamma function in a manner he described as both sketchy and overly complicated. Author Emil Artin, one of the twentieth century's leading mathematicians, wrote in his Preface to this book, "I feel that this monograph will help to show that the gamma function can be thought of as one of the elementary functions, and that all of its basic properties can be established using elementary methods of the calculus." Generations of teachers
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Ignacio Zalduendo · Springer undergraduate mathematics series · 2022
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William B. Jordan · SIAM Review · 1983
Previous article Next article Gamma Function ExpansionsW. B. JordanW. B. Jordanhttps://doi.org/10.1137/1025053PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Gamma Function Expansions." SIAM Review, 25(2), p. 268 Previous article Next article FiguresRelatedReferencesCited byDetails Volume 25, Issue 2| 1983SIAM Review History Published online:17 February 2012 InformationCopyright © 1983 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1025053Article page range:pp. 268-268ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics
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