Gelfond's Constant

Author

Eric W. Weisstein
March 21, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/GelfondsConstant.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Digits

N[Exp[Pi],20]

RealDigits[Exp[Pi],10,10]

Continued Fraction

ContinuedFraction[Exp[Pi],50]

Transcendence

Element[E^Pi,Algebraics]

Iterations

RSolve[{k[n]==((1-(Sqrt[1-((k[n-1])^2)]))/(1+(Sqrt[1-((k[n-1])^2)]))),k[0]==(1/(Sqrt[2]))},k[n],n]

k[n_]:=k[n]=((1-(Sqrt[1-((k[n-1])^2)]))/(1+(Sqrt[1-((k[n-1])^2)])))
k[0]:=1/Sqrt[2]

FullSimplify[RootReduce[k/@Range[0,3]]]

N[Table[(((((k[n])/4)))^(-1/(2^(n-1))))-(E^π),{n,4}],20]

N[Table[(((((k[n])/4)))^(-(2^(1-n))))-(E^π),{n,5}],20]