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Euler-mascheroni Constant
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:\gamma = \lim_{n ightarrow \infty } \left( \left(
\sum_{k=1}^n rac{1}{k} ight) - \ln(n) ight)=\int_1^\infty\left({1\over\lfloor x floor}-{1\over x} ight)\,dx

Its approximate value is γ ≈ 0.57721 56649 01532 86060
65120 90082 40243 10421 59335


HISTORY

The constant was first defined by Swiss mathematician Leonhard Euler in a paper ''De Progressionibus harmonicus observationes'' published in 1735 . Euler used the notation ''C'' for the constant, and initially calculated its value to 6 decimal places. In 1761 he extended this calculation, publishing a value to 16 decimal places. In 1790 Italian mathematician Lorenzo Mascheroni introduced the notation γ for the constant, and attempted to extend Euler's calculation still further, to 32 decimal places, although subsequent calculations showed that he had made an error in the 20th decimal place.

It is not known whether γ is a Rational Number or not. However, Continued Fraction analysis shows that if γ is rational, its denominator has more than 10242080 digits (Havil, page 97).


PROPERTIES

The constant is given by several Integral s:

:\gamma = - \int_0^\infty { e^{-x} \ln(x) }\,dx

:: = - \int_0^1 { \ln\ln\left ( rac{1}{x} ight ) }\,dx

:: = \int_0^\infty {\left ( rac{1}{1-e^{-x}}- rac{1}{x} ight )e^{-x} }\,dx

:: = \int_0^\infty { rac{1}{x} \left ( rac{1}{1+x}-e^{-x} ight ) }\,dx.

Other integrals that include \gamma are:

: \int_0^\infty { e^{-x^2} \ln(x) }\,dx = -1/4(\gamma+2 \ln2) \sqrt{\pi}

: \int_0^\infty { e^{-x} (\ln(x))^2 }\,dx = \gamma^2 +1/6 \pi^2 .

One can express \gamma as a Double Integral also:

: \gamma = \int_{0}^{1}\int_{0}^{1} rac{x-1}{(1-x\,y)\ln(x\,y)} \, dx\,dy.

An interesting comparison by J. Sondow (2005) is the double integral

: \ln \left ( rac{4}{\pi} ight ) = \int_{0}^{1}\int_{0}^{1} rac{x-1}{(1+x\,y)\ln(x\,y)} \, dx\,dy.

It shows that \ln \left ( rac{4}{\pi} ight ) may be thought of an "alternating Euler constant".

In 1910, Vacca gave the interesting sum

: \gamma = \sum_{m=1}^\infty (-1)^m rac{ \left \lfloor \log_2 m ight floor}{m}

where \log_2 is the Logarithm of base 2 and \left \lfloor \, ight floor is the Floor Function .

Vacca's series may be obtained by manipulation of Catalan's integral

: \gamma = \int_0^1 rac{1}{1+x} \sum_{n=1}^\infty x^{2^n-1} \, dx.


RELATIONS TO SPECIAL FUNCTIONS


\gamma can also be expressed as an Infinite Sum with terms involving the values of the Riemann Zeta Function at positive integers:

:\gamma = \sum_{m=2}^{\infty} rac{(-1)^m\zeta(m)}{m}

:= \ln \left ( rac{4}{\pi} ight ) + \sum_{m=1}^{\infty} rac{(-1)^{m-1} \zeta(m+1)}{2^m (m+1)}.

Other Zeta-related series include

: \gamma = rac{3}{2}- \ln 2 - \sum_{m=2}^\infty (-1)^m\, rac{m-1}{m} {Link without Title}

:: = \lim_{n o \infty} \left [ rac{2\,n-1}{2\,n} - \ln\,n + \sum_{k=2}^n \left ( rac{1}{k} - rac{\zeta(1-k)}{n^k} ight ) ight ].

:: = \lim_{n o \infty} \left [ rac{2^n}{e^{2^n}} \sum_{m=0}^\infty rac{2^{m \,n}}{(m+1)!} \sum_{t=0}^m rac{1}{t+1} - n\, \ln2+ O \left ( rac{1}{2^n\,e^{2^n}} ight ) ight ]

The error term in last identity is a rapidly decreasing function of ''n''. As a result, the formula is well-suited to efficiently computing the constant to high precision.

A limit related to the Beta Function (in terms of Gamma Function s) is

: \gamma = \lim_{n o \infty} \left [ rac{ \Gamma( rac{1}{n}) \Gamma(n+1)\, n^{1+1/n}}{\Gamma(2+n+ rac{1}{n})} - rac{n^2}{n+1} ight ].

Two other interesting limits equaling the Euler-Mascheroni constant are the antisymmetric limit

: \gamma = \lim_{s o 1} \sum_{n=1}^\infty \left ( rac{1}{n^s}- rac{1}{s^n} ight )

and

: \gamma = \lim_{x o \infty} \left x - \Gamma \left ( rac{1}{x} ight ) ight

:: = \lim_{n o \infty} rac{1}{n}\, \sum_{k=1}^n \left ( \left \lceil rac{n}{k} ight ceil - rac{n}{k} ight ).

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:

:\gamma = \sum_{k=1}^n rac{1}{k} - \ln(n) -
\sum_{m=2}^\infty rac{\zeta (m,n+1)}{m}
where \zeta(s,k) is the Hurwitz Zeta Function . The sum in this equation involves the Harmonic Number s, Hn. Expanding some of the terms in the Hurwitz zeta function gives:

:
H_n = \ln n + \gamma + rac {1} {2n} - rac {1} {12n^2} + rac {1} {120n^4} - arepsilon , where 0 < arepsilon < rac {1} {252n^6}.

There is also the related limit:
:
\gamma = \lim_{n o \infty} (H_{n-1} - \ln n).


The constant can also be calculated as a derivative of Euler's Gamma Function :
:\gamma = -\Gamma'(1).


E TO THE POWER OF &GAMMA;


The constant eγ is also important in number theory. Occasionally, eγ is denoted y' It is expressed with the following Limit , where pn is the ''n''-th Prime Number :

:
e^\gamma = \lim_{n o \infty} rac {1} {\ln p_n} \prod_{i=1}^n rac {p_i} {p_i - 1}

which is a restatement of the third of Mertens' Theorems . The numerical value of eγ is:
:e^\gamma =1.78107241799019798523650410310717954916964521430343\dots

Other Infinite Product s relating to e^{\gamma} include

: rac{e^{1+\gamma /2}}{\sqrt{2\,\pi}} = \prod_{n=1}^\infty e^{-1+1/(2\,n)}\,\left (1+ rac{1}{n} ight )^n

: rac{e^{3+2\gamma}}{2\, \pi} = \prod_{n=1}^\infty e^{-2+2/n}\,\left (1+ rac{2}{n} ight )^n.

Both of these products result from the Barnes G-function

: e^{\gamma} = \left ( rac{2}{1} ight )^{1/2} \left ( rac{2^2}{1 \cdot 3} ight )^{1/3} \left ( rac{2^3 \cdot 4}{1 \cdot 3^3} ight )^{1/4} \cdots

It is due to J. Sondow using Hypergeometric Function s.


APPEARANCES


The Euler-Mascheroni constant appears, among other places, in:


REFERENCES