Euler–Mascheroni constant
The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter Template:Lang (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma ).
It is defined as the limiting difference between the harmonic series and the natural logarithm:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma =\lim _{{n\rightarrow \infty }}\left(\sum _{{k=1}}^{n}{\frac {1}{k}}-\ln(n)\right)=\int _{1}^{\infty }\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx.
Here, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \lfloor x\rfloor represents the floor function.
The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is
- Template:Gaps.^{[1]}
Template:Irrational numbers | |
Binary | Template:Gaps |
Decimal | Template:Gaps |
Hexadecimal | Template:Gaps |
Continued fraction | Template:Nowrap^{[2]} (This continued fraction is not known to be finite or periodic. Shown in linear notation) |
Contents
History
The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations A and a for the constant. The notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time perhaps because of the constant's connection to the gamma function.^{[3]} For example, the German mathematician Carl Anton Bretschneider used the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma in 1835^{[4]} and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.^{[5]}
Appearances
The Euler–Mascheroni constant appears, among other places, in the following ('*' means that this entry contains an explicit equation):
- Expressions involving the exponential integral*
- The Laplace transform* of the natural logarithm
- The first term of the Taylor series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants*
- Calculations of the digamma function
- A product formula for the gamma function
- An inequality for Euler's totient function
- The growth rate of the divisor function
- The calculation of the Meissel–Mertens constant
- The third of Mertens' theorems*
- Solution of the second kind to Bessel's equation
- In the regularization/renormalization of the Harmonic series as a finite value
- In Dimensional regularization of Feynman diagrams in Quantum Field Theory
- The mean of the Gumbel distribution
- The information entropy of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.
- The answer to the coupon collector's problem*
- In some formulations of Zipf's law
- A definition of the cosine integral*
- In expressions revealing the key properties of an exoplanet atmosphere (temperature, pressure, and composition) embedded in its absorption spectrum, which are at the basis of a new method to determine the mass of exoplanets, MassSpec.^{[6]}
Properties
The number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma has not been proved algebraic or transcendental. In fact, it is not even known whether Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma is irrational. Continued fraction analysis reveals that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma is rational, its denominator must be greater than 10^{242080}.^{[7]} The ubiquity of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma revealed by the large number of equations below makes the irrationality of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma a major open question in mathematics. Also see Sondow (2003a).
Relation to gamma function
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma is related to the digamma function Ψ, and hence the derivative of the gamma function Γ, when both functions are evaluated at 1. Thus:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \ -\gamma =\Gamma '(1)=\Psi (1).
This is equal to the limits:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): -\gamma =\lim _{{z\to 0}}\left\{\Gamma (z)-{\frac 1{z}}\right\}=\lim _{{z\to 0}}\left\{\Psi (z)+{\frac 1{z}}\right\}.
Further limit results are (Krämer, 2005):
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \lim _{{z\to 0}}{\frac 1{z}}\left\{{\frac 1{\Gamma (1+z)}}-{\frac 1{\Gamma (1-z)}}\right\}=2\gamma
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \lim _{{z\to 0}}{\frac 1{z}}\left\{{\frac 1{\Psi (1-z)}}-{\frac 1{\Psi (1+z)}}\right\}={\frac {\pi ^{2}}{3\gamma ^{2}}}.
A limit related to the beta function (expressed in terms of gamma functions) is
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma =\lim _{{n\to \infty }}\left\{{\frac {\Gamma ({\frac {1}{n}})\Gamma (n+1)\,n^{{1+1/n}}}{\Gamma (2+n+{\frac {1}{n}})}}-{\frac {n^{2}}{n+1}}\right\}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma =\lim \limits _{{m\to \infty }}\sum _{{k=1}}^{m}{m \choose k}{\frac {(-1)^{k}}{k}}\ln(\Gamma (k+1)).
Relation to the zeta function
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\begin{aligned}\gamma &=\sum _{{m=2}}^{{\infty }}(-1)^{m}{\frac {\zeta (m)}{m}}\\&=\ln \left({\frac {4}{\pi }}\right)+\sum _{{m=2}}^{{\infty }}(-1)^{m}{\frac {\zeta (m)}{2^{{m-1}}m}}.\end{aligned}}
Other series related to the zeta function include:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\begin{aligned}\gamma &={\frac {3}{2}}-\ln 2-\sum _{{m=2}}^{\infty }(-1)^{m}\,{\frac {m-1}{m}}[\zeta (m)-1]\\&=\lim _{{n\to \infty }}\left[{\frac {2\,n-1}{2\,n}}-\ln \,n+\sum _{{k=2}}^{n}\left({\frac {1}{k}}-{\frac {\zeta (1-k)}{n^{k}}}\right)\right]\\&=\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].\end{aligned}}
The error term in the last equation is a rapidly decreasing function of n. As a result, the formula is well-suited for efficient computation of the constant to high precision.
Other interesting limits equaling the Euler–Mascheroni constant are the antisymmetric limit (Sondow, 1998)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \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)=\lim _{{s\to 0}}{\frac {\zeta (1+s)+\zeta (1-s)}{2}}
and de la Vallée-Poussin's formula
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\begin{aligned}\gamma =\lim _{{n\to \infty }}{\frac {1}{n}}\,\sum _{{k=1}}^{n}\left(\left\lceil {\frac {n}{k}}\right\rceil -{\frac {n}{k}}\right).\end{aligned}}
Closely related to this is the rational zeta series expression. By peeling off the first few terms of the series above, one obtains an estimate for the classical series limit:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma =\sum _{{k=1}}^{n}{\frac {1}{k}}-\ln n-\sum _{{m=2}}^{\infty }{\frac {\zeta (m,n+1)}{m}}
where ζ(s,k) is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, H_{n}. Expanding some of the terms in the Hurwitz zeta function gives:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): H_{n}=\ln n+\gamma +{\frac {1}{2n}}-{\frac {1}{12n^{2}}}+{\frac {1}{120n^{4}}}-\varepsilon , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): 0<\varepsilon <{\frac {1}{252n^{6}}}.
Integrals
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma equals the value of a number of definite integrals:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\begin{aligned}\gamma &=-\int _{0}^{\infty }{e^{{-x}}\ln x}\,dx=-4\int _{0}^{\infty }{e^{{-x^{2}}}x\ln x}\,dx\\&=-\int _{0}^{1}\ln \ln \left({\frac {1}{x}}\right)dx\\&=\int _{0}^{\infty }\left({\frac 1{e^{x}-1}}-{\frac 1{xe^{x}}}\right)dx=\int _{0}^{1}\left({\frac 1{\ln x}}+{\frac 1{1-x}}\right)dx\\&=\int _{0}^{\infty }\left({\frac 1{1+x^{k}}}-e^{{-x}}\right){\frac {dx}{x}},\quad k>0\\&=\int _{0}^{1}H_{{x}}dx\end{aligned}}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): H_{{x}} is the fractional Harmonic number.
Definite integrals in which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma appears include:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \int _{0}^{\infty }{e^{{-x^{2}}}\ln x}\,dx=-{\tfrac 14}(\gamma +2\ln 2){\sqrt {\pi }}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \int _{0}^{\infty }{e^{{-x}}\ln ^{2}x}\,dx=\gamma ^{2}+{\frac {\pi ^{2}}{6}}.
One can express Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma using a special case of Hadjicostas's formula as a double integral (Sondow 2003a, 2005) with equivalent series:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \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).
An interesting comparison by J. Sondow (2005) is the double integral and alternating series
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \ln \left({\frac {4}{\pi }}\right)=\int _{{0}}^{{1}}\int _{{0}}^{{1}}{\frac {x-1}{(1+x\,y)\ln(x\,y)}}\,dx\,dy=\sum _{{n=1}}^{\infty }(-1)^{{n-1}}\left({\frac {1}{n}}-\ln {\frac {n+1}{n}}\right).
It shows that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \ln \left({\frac {4}{\pi }}\right) may be thought of as an "alternating Euler constant".
The two constants are also related by the pair of series (see Sondow 2005 #2)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \sum _{{n=1}}^{\infty }{\frac {N_{1}(n)+N_{0}(n)}{2n(2n+1)}}=\gamma
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \sum _{{n=1}}^{\infty }{\frac {N_{1}(n)-N_{0}(n)}{2n(2n+1)}}=\ln \left({\frac {4}{\pi }}\right)
where N_{1}(n) and N_{0}(n) are the number of 1's and 0's, respectively, in the base 2 expansion of n.
We have also Catalan's 1875 integral (see Sondow and Zudilin)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma =\int _{0}^{1}{\frac {1}{1+x}}\sum _{{n=1}}^{\infty }x^{{2^{n}-1}}\,dx.
Series expansions
Euler showed that the following infinite series approaches Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma :
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma =\sum _{{k=1}}^{\infty }\left[{\frac {1}{k}}-\ln \left(1+{\frac {1}{k}}\right)\right].
The series for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma is equivalent to series Nielsen found in 1897:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma =1-\sum _{{k=2}}^{{\infty }}(-1)^{k}{\frac {\lfloor \log _{2}k\rfloor }{k+1}}.
In 1910, Vacca found the closely related series:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\gamma =\sum _{{k=2}}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k}}={\frac 12}-{\frac 13}+2\left({\frac 14}-{\frac 15}+{\frac 16}-{\frac 17}\right)+3\left({\frac 18}-{\frac 19}+{\frac 1{10}}-{\frac 1{11}}+\dots -{\frac 1{15}}\right)+\dots }
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \log _{2} is the logarithm of base 2 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \lfloor \,\rfloor is the floor function.
In 1926 he found a second series:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\gamma +\zeta (2)=\sum _{{k=2}}^{\infty }\left({\frac 1{\lfloor {\sqrt {k}}\rfloor ^{2}}}-{\frac 1{k}}\right)=\sum _{{k=2}}^{{\infty }}{\frac {k-\lfloor {\sqrt {k}}\rfloor ^{2}}{k\lfloor {\sqrt {k}}\rfloor ^{2}}}={\frac 12}+{\frac 23}+{\frac 1{2^{2}}}\sum _{{k=1}}^{{2\times 2}}{\frac k{k+2^{2}}}+{\frac 1{3^{2}}}\sum _{{k=1}}^{{3\times 2}}{\frac k{k+3^{2}}}+\dots }.
From the Malmsten-Kummer-expansion for the logarithm of the gamma function we get:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma =\ln \pi -4\ln \Gamma ({\tfrac 34})+{\frac 4{\pi }}\sum _{{k=1}}^{{\infty }}(-1)^{{k+1}}{\frac {\ln(2k+1)}{2k+1}}.
Series of prime numbers:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\begin{aligned}\gamma =\lim _{{n\to \infty }}\left(\ln n-\sum _{{p\leq n}}{\frac {\ln p}{p-1}}\right)\end{aligned}}. ^{[8]}
Series relating to square roots:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \lim_{n \rightarrow \infty}\left [ \sum_{k=1}^n \frac{1}{k} - \ln \sqrt { \sum_{k=1}^n k } \right ] - \ln \sqrt 2} ^{[9]}
Asymptotic expansions
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma equals the following asymptotic formulas (where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): H_n is the nth harmonic number.)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma \sim H_{n}-\ln \left(n\right)-{\frac {1}{{2n}}}+{\frac {1}{{12n^{2}}}}-{\frac {1}{{120n^{4}}}}+...
- (Euler)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma \sim H_{n}-\ln \left({n+{\frac {1}{2}}+{\frac {1}{{24n}}}-{\frac {1}{{48n^{3}}}}+...}\right)
- (Negoi)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma \sim 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}}}}-...
- (Cesaro)
The third formula is also called the Ramanujan expansion.
Relations with the reciprocal logarithm
The reciprocal logarithm function (Krämer, 2005)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\frac {z}{\ln(1-z)}}=\sum _{{n=0}}^{{\infty }}C_{n}z^{n},\quad |z|<1,
has a deep connection with Euler's constant and was studied by James Gregory in connection with numerical integration. The coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): C_{n} are called Gregory coefficients; the first six were given in a letter to John Collins in 1670. From the equations
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): C_{0}=-1\;,\quad \sum _{{k=0}}^{n}{\frac {C_{k}}{n+1-k}}=0,\quad n=1,2,3,\dots
, which can be used recursively to get these coefficients for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): n\geq 1 , we get the table
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | OEIS sequences |
---|---|---|---|---|---|---|---|---|---|---|---|
C_{n} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\tfrac 12} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\tfrac 1{12}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\tfrac 1{24}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\tfrac {19}{720}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\tfrac 3{160}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\tfrac {863}{60480}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\tfrac {275}{24192}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\tfrac {33953}{3628800}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\tfrac {8183}{1036800}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\tfrac {3250433}{479001600}} | Template:OEIS2C (numerators),
Template:OEIS2C(denominators) |
Gregory coefficients are similar to Bernoulli numbers and satisfy the asymptotic relation
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): C_{n}={\frac 1{n\ln ^{2}n}}-{\mathcal {O}}\left({\frac 1{n\ln ^{3}n}}\right),\quad n\to \infty ,
and the integral representation
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): C_{n}=\int _{0}^{{\infty }}{\frac {dx}{(1+x)^{n}\left(\ln ^{2}x+\pi ^{2}\right)}},\quad n=1,2,\dots .
Euler's constant has the integral representations
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma =\int _{0}^{{\infty }}{\frac {\ln(1+x)}{\ln ^{2}x+\pi ^{2}}}\cdot {\frac {dx}{x^{2}}}=\int _{{-\infty }}^{{\infty }}{\frac {\ln(1+e^{{-x}})}{x^{2}+\pi ^{2}}}\,e^{x}\,dx.
A very important expansion of Gregorio Fontana (1780) is:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\begin{aligned}H_{n}&=\gamma +\log n+{\frac 1{2n}}-\sum _{{k=2}}^{{\infty }}{\frac {(k-1)!C_{k}}{n(n+1)\dots (n+k-1)}},\quad n=1,2,\dots ,\\&=\gamma +\log n+{\frac 1{2n}}-{\frac 1{12n(n+1)}}-{\frac 1{12n(n+1)(n+2)}}-{\frac {19}{120n(n+1)(n+2)(n+3)}}-\dots \end{aligned}}
which is convergent for all n.
Weighted sums of the Gregory coefficients give different constants:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\begin{aligned}1&=\sum _{{n=1}}^{{\infty }}C_{n}={\tfrac 12}+{\tfrac 1{12}}+{\tfrac 1{24}}+{\tfrac {19}{720}}+{\tfrac 3{160}}+\dots ,\\{\frac 1{\log 2}}-1&=\sum _{{n=1}}^{{\infty }}(-1)^{{n+1}}C_{n}={\tfrac 12}-{\tfrac 1{12}}+{\tfrac 1{24}}-{\tfrac {19}{720}}+{\tfrac 3{160}}-\dots ,\\\gamma &=\sum _{{n=1}}^{{\infty }}{\frac {C_{n}}{n}}={\tfrac 12}+{\tfrac 1{24}}+{\tfrac 1{72}}+{\tfrac {19}{2880}}+{\tfrac 3{800}}+\dots .\end{aligned}}
e^{γ}
The constant e^{γ} is important in number theory. Some authors denote this quantity simply as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma ^{\prime } . e^{γ} equals the following limit, where p_{n} is the nth prime number:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): e^{\gamma }=\lim _{{n\to \infty }}{\frac {1}{\ln p_{n}}}\prod _{{i=1}}^{n}{\frac {p_{i}}{p_{i}-1}}.
This restates the third of Mertens' theorems. The numerical value of e^{γ} is:
- 1.78107241799019798523650410310717954916964521430343 … Template:OEIS2C.
Other infinite products relating to e^{γ} include:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\frac {e^{{1+\gamma /2}}}{{\sqrt {2\,\pi }}}}=\prod _{{n=1}}^{\infty }e^{{-1+1/(2\,n)}}\,\left(1+{\frac {1}{n}}\right)^{n}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\frac {e^{{3+2\gamma }}}{2\,\pi }}=\prod _{{n=1}}^{\infty }e^{{-2+2/n}}\,\left(1+{\frac {2}{n}}\right)^{n}.
These products result from the Barnes G-function.
We also have
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): e^{{\gamma }}=\left({\frac {2}{1}}\right)^{{1/2}}\left({\frac {2^{2}}{1\cdot 3}}\right)^{{1/3}}\left({\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}\right)^{{1/4}}\left({\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}\right)^{{1/5}}\cdots
where the nth factor is the (n+1)st root of
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \prod _{{k=0}}^{n}(k+1)^{{(-1)^{{k+1}}{n \choose k}}}.
This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow (2003) using hypergeometric functions.
Continued fraction
The continued fraction expansion of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma is of the form [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] Template:OEIS2C, of which there is no apparent pattern. The continued fraction has at least 470,000 terms,^{[7]} and it has infinitely many terms if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma is irrational.
Generalizations
Euler's generalized constants are given by
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma _{\alpha }=\lim _{{n\to \infty }}\left[\sum _{{k=1}}^{n}{\frac {1}{k^{\alpha }}}-\int _{1}^{n}{\frac {1}{x^{\alpha }}}\,dx\right],
for 0 < α < 1, with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma as the special case α = 1.^{[10]} This can be further generalized to
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): c_{f}=\lim _{{n\to \infty }}\left[\sum _{{k=1}}^{n}f(k)-\int _{1}^{n}f(x)\,dx\right]
for some arbitrary decreasing function f. For example,
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): f_{n}(x)={\frac {\ln ^{n}x}{x}}
gives rise to the Stieltjes constants, and
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): f_{a}(x)=x^{{-a}}
gives
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma _{{f_{a}}}={\frac {(a-1)\zeta (a)-1}{a-1}}
where again the limit
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma =\lim _{{a\to 1}}\left[\zeta (a)-{\frac {1}{a-1}}\right]
appears.
A two-dimensional limit generalization is the Masser–Gramain constant.
Euler-Lehmer constants are given by summation of inverses of numbers in a common modulo class^{[11]} ,^{[12]}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma (a,q)=\lim _{{x\to \infty }}\left(\sum _{{0<n\leq x \atop n\equiv a{\pmod q}}}{\frac {1}{n}}-{\frac {\log x}{q}}\right).
The basic properties are
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma (0,q)={\frac {\gamma -\log q}{q}},
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \sum _{{a=0}}^{{q-1}}\gamma (a,q)=\gamma ,
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): q\gamma (a,q)=\gamma -\sum _{{j=1}}^{{q-1}}e^{{-2\pi aij/q}}\log(1-e^{{2\pi ij/q}}),
and if Template:Math then
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): q\gamma (a,q)={\frac {q}{d}}\gamma (a/d,q/d)-\log d.
Published digits
Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Mascheroni attempted to calculate the constant to 32 decimal places, but made errors in the 20th–22nd decimal places; starting from the 20th digit, he calculated ...1811209008239 when the correct value is ...0651209008240.
Date | Decimal digits | Author |
---|---|---|
1734 | 5 | Leonhard Euler |
1735 | 15 | Leonhard Euler |
1790 | 19 | Lorenzo Mascheroni |
1809 | 22 | Johann G. von Soldner |
1811 | 22 | Carl Friedrich Gauss |
1812 | 40 | Friedrich Bernhard Gottfried Nicolai |
1857 | 34 | Christian Fredrik Lindman |
1861 | 41 | Ludwig Oettinger |
1867 | 49 | William Shanks |
1871 | 99 | James W.L. Glaisher |
1871 | 101 | William Shanks |
1877 | 262 | J. C. Adams |
1952 | 328 | John William Wrench, Jr. |
1961 | 1050 | Helmut Fischer and Karl Zeller |
1962 | 1,271 | Donald Knuth |
1962 | 3,566 | Dura W. Sweeney |
1973 | 4,879 | William A. Beyer and Michael S. Waterman |
1977 | 20,700 | Richard P. Brent |
1980 | 30,100 | Richard P. Brent & Edwin M. McMillan |
1993 | 172,000 | Jonathan Borwein |
2009 | 29,844,489,545 | Alexander J. Yee & Raymond Chan^{[13]} |
2013 | 119,377,958,182 | Alexander J. Yee^{[13]} |
See also
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Notes
- Footnotes
- References
- Template:Cite journal Derives γ as sums over Riemann zeta functions.
- Gourdon, Xavier, and Sebah, P. (2002) "Collection of formulas for Euler's constant, γ."
- Gourdon, Xavier, and Sebah, P. (2004) "The Euler constant: γ."
- Donald Knuth (1997) The Art of Computer Programming, Vol. 1, 3rd ed. Addison-Wesley. ISBN 0-201-89683-4
- 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.
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- Sondow, Jonathan (2003a) "Criteria for irrationality of Euler's constant," Proceedings of the American Mathematical Society 131: 3335-3344.
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- Sondow, Jonathan (2005) "New Vacca-type rational series for Euler's constant and its 'alternating' analog ln 4/π."
- Template:Cite arXiv Ramanujan Journal 12: 225-244.
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- James Whitbread Lee Glaisher (1872), "On the history of Euler's constant". Messenger of Mathematics vol.1, p. 25-30, JFM 03.0130.01
- Template:Cite journal (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
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External links
- Template:Mathworld
- Jonathan Sondow.
- Fast Algorithms and the FEE Method, E.A. Karatsuba (2005)
- Further formulae which make use of the constant: Gourdon and Sebah (2004).
- ↑ Template:OEIS2C
- ↑ Template:OEIS2C
- ↑ Template:Cite journal
- ↑ Carl Anton Bretschneider: Theoriae logarithmi integralis lineamenta nova (13 October 1835), Journal für die reine und angewandte Mathematik 17, 1837, pp. 257–285 (in Latin; "γ = c = 0,577215 664901 532860 618112 090082 3.." on p. 260)
- ↑ Augustus De Morgan: The differential and integral calculus, Baldwin and Craddock, London 1836–1842 ("γ" on p. 578)
- ↑ Template:Cite journal
- ↑ ^{7.0} ^{7.1} Havil 2003 p 97.
- ↑ http://mathworld.wolfram.com/MertensConstant.html (15)
- ↑ http://mathworld.wolfram.com/Euler-MascheroniConstant.html
- ↑ Havil, 117-118
- ↑ Template:Cite journal
- ↑ Template:Cite journal
- ↑ ^{13.0} ^{13.1} Nagisa – Large Computations