提示:此条目的主题不是
尤拉數。
歐拉-馬斯刻若尼常數歐拉-馬斯刻若尼常數 |
---|
![](//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Gamma-area.svg/220px-Gamma-area.svg.png) 藍色區域的面積收斂到歐拉常數 |
識別 |
---|
符號 | ![{\displaystyle \gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a) |
---|
位數數列編號 | A001620 |
---|
性質 |
---|
定義 | ![{\displaystyle \gamma =\lim _{n\rightarrow \infty }\left[\left(\sum _{k=1}^{n}{\frac {1}{k}}\right)-\ln(n)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7af12010d95e43a1b424a6c3a76c92c2727c1c06)
![{\displaystyle \gamma =\int _{1}^{\infty }\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dfbeb7f78ec31260e483c0343ff28b1ea8054b6) |
---|
連分數 | [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] |
---|
表示方式 |
---|
值 | 0.57721566490153... |
---|
無窮級數 | ![{\displaystyle \gamma =\sum _{k=1}^{\infty }\left[{\frac {1}{k}}-\ln \left(1+{\frac {1}{k}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7176d3dc104a5ef96f1039306f330096dde13ccd) |
---|
|
二进制 | 0.100100111100010001100111… |
---|
十进制 | 0.577215664901532860606512… |
---|
十六进制 | 0.93C467E37DB0C7A4D1BE3F81… |
---|
|
|
歐拉-馬斯刻若尼常數是一个数学常数,定义为调和级数与自然对数的差值:
![{\displaystyle \gamma =\lim _{n\rightarrow \infty }\left[\left(\sum _{k=1}^{n}{\frac {1}{k}}\right)-\ln(n)\right]=\int _{1}^{\infty }\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf185b6d9ad97389a24d868420ea2379a75f60a)
它的近似值为
[1],
歐拉-馬斯刻若尼常數主要应用于数论。
该常数最先由瑞士数学家莱昂哈德·欧拉在1735年发表的文章De Progressionibus harmonicus observationes中定义。欧拉曾经使用
作为它的符号,并计算出了它的前6位小数。1761年他又将该值计算到了16位小数。1790年,意大利数学家洛倫佐·馬斯凱羅尼引入了
作为这个常数的符号,并将该常数计算到小数点后32位。但后来的计算显示他在第20位的时候出现了错误。
目前尚不知道该常数是否为有理数,但是分析表明如果它是一个有理数,那么它的分母位数将超过10242080。[2]
与伽玛函数的关系[编辑]
。
。
。
与ζ函数的关系[编辑]
![{\displaystyle \gamma =\sum _{m=2}^{\infty }{\frac {(-1)^{m}\zeta (m)}{m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7b3ca3c0658e49949d993f9c0ccca171c368fd)
。
![{\displaystyle \lim _{\varepsilon \to 0}{\frac {\zeta (1+\varepsilon )+\zeta (1-\varepsilon )}{2}}=\gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8953c45930358e67ed50421c45f50cb88a1766)
![{\displaystyle \gamma ={\frac {3}{2}}-\ln 2-\sum _{m=2}^{\infty }(-1)^{m}\,{\frac {m-1}{m}}[\zeta (m)-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87c6c8e2b81d1f232a24da48013a831eb1e9192d)
。
![{\displaystyle =\lim _{n\to \infty }\left[{\frac {2^{n}}{e^{2^{n}}}}\sum _{m=0}^{\infty }{\frac {2^{m\,n}}{(m+1)!}}\sum _{t=0}^{m}{\frac {1}{t+1}}-n\,\ln 2+O\left({\frac {1}{2^{n}\,e^{2^{n}}}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68a0383cf4feb0933d76bad33d30d5077a663d96)
![{\displaystyle \gamma =\lim _{s\to 1^{+}}\sum _{n=1}^{\infty }\left({\frac {1}{n^{s}}}-{\frac {1}{s^{n}}}\right)=\lim _{s\to 1^{+}}\left(\zeta (s)-{\frac {1}{s-1}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4258cbf97ed808b62c3ecc60d95d10fd899a7e)
![{\displaystyle \gamma =\lim _{x\to \infty }\left[x-\Gamma \left({\frac {1}{x}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6598511fab1a65e9b06953c415d96ee8633c8f2)
。
![{\displaystyle \gamma =\sum _{k=1}^{n}{\frac {1}{k}}-\ln(n)-\sum _{m=2}^{\infty }{\frac {\zeta (m,n+1)}{m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/933d6aa3cf61fb5a9d2d2d352f1e65d42e433bbd)
[證明 1]![{\displaystyle =-\int _{0}^{1}{\ln \ln {\frac {1}{x}}}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fcb72a18f38408ef4c9e3338e5421e63bc057a0)
![{\displaystyle =\int _{0}^{\infty }{\left({\frac {1}{1-e^{-x}}}-{\frac {1}{x}}\right)e^{-x}}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88a0c98752f9786128aa395ea163bbb161b0325a)
![{\displaystyle =\int _{0}^{\infty }{{\frac {1}{x}}\left({\frac {1}{1+x}}-e^{-x}\right)}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98b02b16e4a73502889226a042070950ae90d3cf)
![{\displaystyle \int _{0}^{\infty }{e^{-x^{2}}\ln x}\,dx=-{\tfrac {1}{4}}(\gamma +2\ln 2){\sqrt {\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d20d929e1f09c903d83085d93ac5c75102b08ad)
。
![{\displaystyle \gamma =\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1-x\,y)\ln(x\,y)}}\,dx\,dy=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\ln {\frac {n+1}{n}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1e9e85289cc42a645d5c6c444ca7784317668a)
![{\displaystyle \sum _{n=1}^{\infty }{\frac {N_{1}(n)+N_{0}(n)}{2n(2n+1)}}=\gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce24aac17a79472d56c1e8188deac5adde35bb09)
级数展开式[编辑]
![{\displaystyle \gamma =\sum _{k=1}^{\infty }\left[{\frac {1}{k}}-\ln \left(1+{\frac {1}{k}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7176d3dc104a5ef96f1039306f330096dde13ccd)
.
![{\displaystyle \gamma =\sum _{k=2}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k}}={\tfrac {1}{2}}-{\tfrac {1}{3}}+2\left({\tfrac {1}{4}}-{\tfrac {1}{5}}+{\tfrac {1}{6}}-{\tfrac {1}{7}}\right)+3\left({\tfrac {1}{8}}-\dots -{\tfrac {1}{15}}\right)+\dots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c0dda40fa4ecfde8adc649afb910bbee619668)
![{\displaystyle \gamma =\int _{0}^{1}{\frac {1}{1+x}}\sum _{n=1}^{\infty }x^{2^{n}-1}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31e3d5a0521b8281008b33e1f6f44e34f2e98010)
的连分数展开式为:
(OEIS數列A002852).
渐近展开式[编辑]
![{\displaystyle \gamma \approx H_{n}-\ln \left(n\right)-{\frac {1}{2n}}+{\frac {1}{12n^{2}}}-{\frac {1}{120n^{4}}}+...}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86d001fe31b0f0a1ba644990d4d0586ac2656ab4)
![{\displaystyle \gamma \approx H_{n}-\ln \left({n+{\frac {1}{2}}+{\frac {1}{24n}}-{\frac {1}{48n^{3}}}+...}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b388e0a5f0d1c2c06e001a4d41320610c7bb42c1)
![{\displaystyle \gamma \approx H_{n}-{\frac {\ln \left(n\right)+\ln \left({n+1}\right)}{2}}-{\frac {1}{6n\left({n+1}\right)}}+{\frac {1}{30n^{2}\left({n+1}\right)^{2}}}-...}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d32363e50b0f87916f0ab1bd836fede017ddfcb2)
已知位数[编辑]
的已知位数 日期 | 位数 | 计算者 |
1734年 | 5 | 莱昂哈德·欧拉 |
1736年 | 15 | 莱昂哈德·欧拉 |
1790年 | 19 | 洛倫佐·馬斯凱羅尼 |
1809年 | 24 | Johann G. von Soldner |
1812年 | 40 | F.B.G. Nicolai |
1861年 | 41 | Oettinger |
1869年 | 59 | William Shanks |
1871年 | 110 | William Shanks |
1878年 | 263 | 约翰·柯西·亚当斯 |
1962年 | 1,271 | 高德纳 |
1962年 | 3,566 | D.W. Sweeney |
1977年 | 20,700 | Richard P. Brent |
1980年 | 30,100 | Richard P. Brent和埃德温·麦克米伦 |
1993年 | 172,000 | Jonathan Borwein |
1997年 | 1,000,000 | Thomas Papanikolaou |
1998年12月 | 7,286,255 | Xavier Gourdon |
1999年10月 | 108,000,000 | Xavier Gourdon和Patrick Demichel |
2006年7月16日 | 2,000,000,000 | Shigeru Kondo和Steve Pagliarulo |
2006年12月8日 | 116,580,041 | Alexander J. Yee |
2007年7月15日 | 5,000,000,000 | Shigeru Kondo和Steve Pagliarulo |
2008年1月1日 | 1,001,262,777 | Richard B. Kreckel |
2008年1月3日 | 131,151,000 | Nicholas D. Farrer |
2008年6月30日 | 10,000,000,000 | Shigeru Kondo和Steve Pagliarulo |
2009年1月18日 | 14,922,244,771 | Alexander J. Yee和Raymond Chan |
2009年3月13日 | 29,844,489,545 | Alexander J. Yee和Raymond Chan |
2013年 | 119,377,958,182 | Alexander J. Yee |
2016年 | 160,000,000,000 | Peter Trueb |
2016年 | 250,000,000,000 | Ron Watkins |
2017年 | 477,511,832,674 | Ron Watkins |
2020年 | 600,000,000,100 | Seungmin Kim和Ian Cutress |
相关证明[编辑]
- ^
的证明:
首先根据放缩法(
)容易知道,
,以及
。因此
存在并有限。
![{\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f62f9dd76fe1b16dd3bb939a35457920156a8963)
![{\displaystyle =\sum _{k=1}^{n}\int _{0}^{1}t^{k-1}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6be1619c9b832319ca54b0ebf9063b002ba21ce)
![{\displaystyle =\int _{0}^{1}\sum _{k=1}^{n}t^{k-1}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd86128876a9a70c99a03de498c171148c23b570)
![{\displaystyle =\int _{0}^{1}{\frac {1-t^{n}}{1-t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce90954d48b9984d33ef59d558be141f07b54843)
![{\displaystyle =\int _{n}^{0}{\frac {1-\left(1-{\frac {x}{n}}\right)^{n}}{1-\left(1-{\frac {x}{n}}\right)}}d\left(1-{\tfrac {x}{n}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e29e6b8ca90ffde85332cd87268bd20064499737)
![{\displaystyle =\int _{n}^{0}{\frac {1-\left(1-{\frac {x}{n}}\right)^{n}}{\frac {x}{n}}}\left(-{\frac {1}{n}}\right)dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/332d4a9bca716f67214c6fdd8294b4b176f73c9d)
![{\displaystyle =\int _{0}^{n}{\frac {1-\left(1-{\frac {x}{n}}\right)^{n}}{x}}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20749b2b31b10bea5f753f4bd929730f6cbf4e4b)
而![{\displaystyle \ln n=\int _{1}^{n}{\frac {1}{x}}\,dx,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eba5bb209ddce67998c0be7dc78b3b40e3230723)
所以![{\displaystyle \gamma =\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln n\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f61eb7f7c563396a9a8abb1725b4355cfa398d6b)
![{\displaystyle =\lim _{n\to \infty }\left[\int _{0}^{n}{\frac {1-(1-x/n)^{n}}{x}}\,dx-\int _{1}^{n}{\frac {1}{x}}\,dx\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad51b5b4b416e2d6f007f02e8c37f90b3f3ea93)
![{\displaystyle =\lim _{n\to \infty }\left[\int _{0}^{1}{\frac {1-(1-x/n)^{n}}{x}}\,dx-\int _{1}^{n}{\frac {(1-x/n)^{n}}{x}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f0df6992c0f206f2e4328fca6798cb2fa271768)
(单调收敛定理)
![{\displaystyle =\int _{0}^{1}{\frac {1-e^{-x}}{x}}\,dx-\int _{1}^{\infty }{\frac {e^{-x}}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1419d1d33cd9d6cff3c7c54cea49e7d2de725b)
![{\displaystyle =\left.(1-e^{-x})\ln x\right|_{0}^{1}-\int _{0}^{1}\ln x\,d(1-e^{-x})-\left.e^{-x}\ln x\right|_{1}^{\infty }+\int _{1}^{\infty }\ln x\,de^{-x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f40467367455e379c341c506716d6b0a6147c38)
前面的放缩法主要是证明了
是单调递减并下有界限(0),所有极限存在。放缩法的结论需要使用ln(1+x)和ln(1-x)的泰勒级数展开进行证明。
參考文獻[编辑]
- ^ A001620 oeis.org [2014-7-17]
- ^ Havil 2003 p 97.
- Borwein, Jonathan M., David M. Bradley, Richard E. Crandall. Computational Strategies for the Riemann Zeta Function (PDF). Journal of Computational and Applied Mathematics. 2000, 121: 11 [2014-07-17]. doi:10.1016/s0377-0427(00)00336-8. (原始内容 (PDF)存档于2006-09-25). Derives γ as sums over Riemann zeta functions.
- Gourdon, Xavier, and Sebah, P. (2002) "Collection of formulas for Euler's constant, γ. (页面存档备份,存于互联网档案馆)"
- Gourdon, Xavier, and Sebah, P. (2004) "The Euler constant: γ. (页面存档备份,存于互联网档案馆)"
- Donald Knuth (1997) The Art of Computer Programming, Vol. 1, 3rd ed. Addison-Wesley. ISBN 978-0-201-89683-1
- Krämer, Stefan (2005) Die Eulersche Konstante γ und verwandte Zahlen. Diplomarbeit, Universität Göttingen.
- Sondow, Jonathan (1998) "An antisymmetric formula for Euler's constant," Mathematics Magazine 71: 219-220.
- Sondow, Jonathan (2002) "A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant." With an Appendix by Sergey Zlobin, Mathematica Slovaca 59: 307-314.
- Sondow, Jonathan. An infinite product for eγ via hypergeometric formulas for Euler's constant, γ. 2003. arXiv:math.CA/0306008
. - Sondow, Jonathan (2003a) "Criteria for irrationality of Euler's constant," Proceedings of the American Mathematical Society 131: 3335-3344.
- Sondow, Jonathan (2005) "Double integrals for Euler's constant and ln 4/π and an analog of Hadjicostas's formula," American Mathematical Monthly 112: 61-65.
- Sondow, Jonathan (2005) "New Vacca-type rational series for Euler's constant and its 'alternating' analog ln 4/π."
- Sondow, Jonathan; Zudilin, Wadim. Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper. 2006. arXiv:math.NT/0304021
. Ramanujan Journal 12: 225-244. - G. Vacca (1926), "Nuova serie per la costante di Eulero, C = 0,577…". Rendiconti, Accademia Nazionale dei Lincei, Roma, Classe di Scienze Fisiche, Matematiche e Naturali (6) 3, 19–20.
- James Whitbread Lee Glaisher (1872), "On the history of Euler's constant". Messenger of Mathematics. New Series, vol.1, p. 25-30, JFM 03.0130.01
- Carl Anton Bretschneider (1837). "Theoriae logarithmi integralis lineamenta nova". Crelle Journal, vol.17, p. 257-285 (submitted 1835)
- Lorenzo Mascheroni (1790). "Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur". Galeati, Ticini.
- Lorenzo Mascheroni (1792). "Adnotationes ad calculum integralem Euleri. In quibus nonnullae formulae ab Eulero propositae evolvuntur". Galeati, Ticini. Both online at: http://books.google.de/books?id=XkgDAAAAQAAJ (页面存档备份,存于互联网档案馆)
- Havil, Julian. Gamma: Exploring Euler's Constant. Princeton University Press. 2003. ISBN 0-691-09983-9.
- Karatsuba, E. A. Fast evaluation of transcendental functions. Probl. Inf. Transm. 1991, 27 (44): 339–360.
- E.A. Karatsuba, On the computation of the Euler constant γ, J. of Numerical Algorithms Vol.24, No.1-2, pp. 83–97 (2000)
- M. Lerch, Expressions nouvelles de la constante d'Euler. Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften 42, 5 p. (1897)
- Lagarias, Jeffrey C. Euler's constant: Euler's work and modern developments. arXiv:1303.1856
. , Bulletin of the American Mathematical Society 50 (4): 527-628 (2013)
外部連結[编辑]