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Euler-Mascheroni Constant Approximations

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A beautiful approximation to the Euler-Mascheroni constant gamma is given by

pi/(2e)=0.57786367...
(1)

(OEIS A086056; E. W. Weisstein, Apr. 18, 2006), which is good to three decimal digits.

In 1982-1983, Odena gave the strange approximation

(0.11111111)^(1/4)=0.5773502677...,
(2)

which is effectively

3^(-1/2)=0.5773502692...
(3)

(Munroe 2012).

Castellanos (1988) gave

(7/(83))^(2/9)=0.57721521...
(4)
((520^2+22)/(52^4))^(1/6)=0.5772156634...
(5)
((80^3+92)/(61^4))^(1/6)=0.57721566457...
(6)
(990^3-55^3-79^2-4^2)/(70^5)=(30316449)/(52521875)=0.5772156649015291...,
(7)

which are good to 6, 8, 9, 14, and 14 digits, respectively.

An approximation involving unit fractions due to P. Galliani (pers. comm., April 1, 2002) is given by

1/2+1/(23)+1/(37)+1/(149)-1/(968625)=0.5772156649012...,
(8)

which differs from gamma by 2.4×10^(-13), i.e., is good to 12 digits.

Ed Pegg, Jr. (pers. comm., March 2, 2002) found

gamma approx 1/(15)+((35)/(263))^(1/3),
(9)

which is good to 8 digits.

M. Hudson (pers. comm., Sept. 3, 2004) found the approximations

gamma approx (0.111)^(1/4)
(10)
approx phi-(51)/(49)
(11)
approx 0.1+(3/(254))^(1/6)
(12)
approx (2/(2533))^(1/13)
(13)
approx sqrt(6/(13))-(19)/(186)
(14)
approx 1/(sqrt(3))-1/(7429)
(15)
approx sqrt((92)/(2025))ln15,
(16)

where phi is the golden ratio, which are good to 5, 5, 6, 7, 7, 8, and 8 digits, respectively.

G. W. Barbosa (pers. comm., Mar. 26 and Apr. 2, 2007) gave

gamma=1-tanh(ln1.57)-(0.57)/(9!)
(17)
=(2(3^0+9)^4)/(8!-5671)-(48+9)/((sqrt(2sqrt(3sqrt(5sqrt(70)))))^(16))
(18)
=(241919341669)/(419114304000),
(19)

which are good to 10 decimal digits, and where the second approximation is a difference of two pandigital parts. Barbosa (pers. comm., Jan. 7, 2008) also gave the pandigital approximation

gamma approx -(e^(-6^3/9))/e+(exp(-exp(e^(.8)))+.4)/(ln2)+(ln5)/(10^7)
(20)

which is good to 13 decimal digits.

See also

Almost Integer, Euler-Mascheroni Constant

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References

Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988a.Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 61, 148-163, 1988b.Friedman, E. "Problem of the Month (August 2004)." https://erich-friedman.github.io/mathmagic/0804.html.Munroe, R. "A Table of Slightly Wrong Equations and Identities Useful for Approximations and/or Trolling Teachers." xkcd: A Webcomic of Romance, Sarcasm, Math, and Language. http://xkcd.com/1047/. Apr. 2012.Sloane, N. J. A. Sequences A086056 in "The On-Line Encyclopedia of Integer Sequences."

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Euler-Mascheroni Constant Approximations

Cite this as:

Weisstein, Eric W. "Euler-Mascheroni Constant Approximations." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Euler-MascheroniConstantApproximations.html

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