Goh-Schmutz Constant

Author

Eric W. Weisstein
February 3, 2015

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/Goh-SchmutzConstant.nb.

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

©2015 Wolfram Research, Inc. except for portions noted otherwise

Value

Sums

-NSum[(E^k/k)*ExpIntegralEi[-k], {k, Infinity}]

Numeric

-Sum[(E^k/k)*ExpIntegralEi[-k], {k, 10^4}]//N

-NSum[(E^k/k)*ExpIntegralEi[-k], {k, Infinity}]

V7

-Sum[(E^k/k)*ExpIntegralEi[-k], {k, Infinity}]

V10

-Sum[(E^k/k)*ExpIntegralEi[-k], {k, Infinity}]

Series

Plot[Log[t + 1]/(E^t - 1),{t,0,10}]

Series[Log[t + 1]/(E^t - 1),{t,0,10}]

SeriesCoefficient[Log[t + 1]/(E^t - 1),{t,0,n}]//Timing

Integrals

(∫_0^∞)((ln(t+1))/((e^t)-1))dt

Integrate[Log[t + 1]/(E^t - 1), {t, 0, Infinity}]

NIntegrate[Log[t + 1]/(E^t - 1), {t, 0, Infinity}]

int1=NIntegrate[Log[1 + t]/(-1 + E^t),{t,0,Infinity},WorkingPrecision->1500,PrecisionGoal->1000,MaxRecursion->16,Method->"DoubleExponential"]

(∫_0^∞)((ln(t+1))/(sinh(t)+cosh(t)-1))dt

ExpToTrig[Log[t + 1]/(E^t - 1)]

Integrate[((Log[1+t])/(-1+Cosh[t]+Sinh[t])), {t, 0, Infinity}]

NIntegrate[((Log[1+t])/(-1+Cosh[t]+Sinh[t])), {t, 0, Infinity}]

(∫_0^∞)ln(1-ln(1-(e^(-t))))dt

Integrate[Log[1 - Log[1 - E^(-t)]], {t, 0, Infinity}]

NIntegrate[Log[1 - Log[1 - E^(-t)]], {t, 0, Infinity}]

int2=NIntegrate[Log[1 + t]/(-1 + E^t),{t,0,Infinity},WorkingPrecision->1500,PrecisionGoal->1000,MaxRecursion->20,Method->"GaussKronrodRule"]

(∫_0^∞)((x exp(-x))/((1-exp(-x)) (1-ln(1-exp(-x)))))dx

Integrate[((x Exp[-x])/((1-Exp[-x])(1-Log[1-Exp[-x]]))),{x,0,∞}]

NIntegrate[((x Exp[-x])/((1-Exp[-x])(1-Log[1-Exp[-x]]))),{x,0,∞}]

Value Comparisons

int1-int2//N