Euler-Mascheroni Constant Approximations

Author

Eric W. Weisstein
March 23, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/Euler-MascheroniConstantApproximations.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Euler-MascheroniConstantApproximations.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Approximations

Barbosa

1-tanh(ln 1.57)-0.57/(9!)

Gerson W. Barbosa
26 Mar 2007

'1-tanh(ln 1.57)-0.57/(9!)'

N[1-Tanh[Log[1.57]]-0.57/9!,10]

CorrectDigits[EulerGamma,1-Tanh[Log[1.57]]-0.57/9!]

CorrectDigits[EulerGamma,expr1=1-Tanh[Log[157/100]]-57/100/9!]

expr1

2*(3^0+9)^4/(8!-5671) - (48+9)/sqrt(2*sqrt(3*sqrt(5*sqrt(70))))^16

Gerson W. Barbosa
2 Apr 2007

((2 ((((3^0)+9))^4))/(8!-5671))-((48+9)/((((Sqrt[2 (Sqrt[3 (Sqrt[5 (Sqrt[70])])])])))^16))

expr2=2*(3^0+9)^4/(8!-5671) - (48+9)/Sqrt[2*Sqrt[3*Sqrt[5*Sqrt[70]]]]^16

expr1==expr2

N[%,10]

CorrectDigits[2*(3^0+9)^4/(8!-5671) - (48+9)/Sqrt[2*Sqrt[3*Sqrt[5*Sqrt[70]]]]^16,EulerGamma]

Pandigital:

(.4 + EXP (-(EXP (EXP (.8)))))/LN (2) + 10^(-7)*LN (5) - EXP (-(6^3/9))/e

-((exp(-((6^3)/9)))/E)+((exp(-exp(exp(.8)))+.4)/(log(2)))+((log(5))/(10^7))

expr=(.4+Exp[-Exp[Exp[.8]]])/Log[2]+10^(-7)*Log[5]-Exp[-(6^3/9)]/E

CorrectDigits[expr,EulerGamma]

expr=(4/10+Exp[-Exp[Exp[8/10]]])/Log[2]+10^(-7)*Log[5]-Exp[-(6^3/9)]/E

CorrectDigits[expr,EulerGamma]

Castellanos

CorrectDigits[{(7/83)^(2/9),EulerGamma}]

CorrectDigits[{((520^2+22)/52^4)^(1/6),EulerGamma}]

CorrectDigits[{((80^3+92)/61^4)^(1/6),EulerGamma}]

CorrectDigits[{(990^3-55^3-79^2-4^2)/70^5,EulerGamma}]

CorrectDigits[{(990^3-55^3-79^2-4^2)/70^5,EulerGamma}]

Galliani

April 1, 2002

1/2+1/23+1/37+1/149-1/968625

CorrectDigits[{%,EulerGamma}]

Hudson

Sept. 3, 2004

approx={
(0.111^(1/4)),GoldenRatio-(51/49),0.1+((((3/254)))^(1/6)),((((2/2533)))^(1/13)),(Sqrt[(6/13)])-(19/186),(1/(Sqrt[3]))-(1/7429),(Sqrt[(92/2025)])Log[15]
};

Ordering[Abs[approx-EulerGamma],All,Greater]

CorrectDigits[{#,EulerGamma}]&/@approx

Odena

N[0.11111111^(1/4)]

CorrectDigits[{0.11111111^(1/4),EulerGamma}]

Pegg

March 2, 2002

CorrectDigits[{(1/15)+((((35/263)))^(1/3)),EulerGamma}]

Weisstein

CorrectDigits[(π/(2E)),EulerGamma]