OFFSET
0,5
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Doctor Bombelli, The Math Forum, Exp[ -I * Pi] = Cos[Pi] + i Sin[Pi] = -1.
FORMULA
a(n) = (-1)^(I Pi) = exp(-Pi)^Pi = (exp( -(1/2)*Pi))^(2*Pi) = sqrt(exp( -Pi)^Pi/(exp(Pi)^Pi)) = exp[(-1/2*Pi)]^(Gamma[1/6]*Gamma[5/6]) = 1/(sqrt[exp[Pi]^(2*Pi)]) = (exp[ -(1/2)*Pi])^(2*Pi) = exp[ -Pi^2].
EXAMPLE
(-1)^(I*Pi) = exp(-Pi)^(Pi) = 0.000051723186...
MATHEMATICA
N[(-1)^(I Pi), 1000] FullSimplify[(-1)^(I Pi) == Exp[ -Pi]^Pi == (Exp[ -(1/2)*Pi])^(2*Pi) == Sqrt[Exp[ -Pi]^Pi/(Exp[Pi]^Pi)] == Exp[(-1/2*Pi)]^(Gamma[1/6]*Gamma[5/6]) == 1/(Sqrt[Exp[Pi]^(2*Pi)]) == (Exp[ -(1/2)*Pi])^(2*Pi) == Exp[ -Pi^2]]
Join[{0, 0, 0, 0}, RealDigits[(Exp[-Pi])^(Pi), 10, 96][[1]]] (* G. C. Greubel, Oct 18 2016 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Marvin Ray Burns, Feb 22 2008, Feb 23 2008
EXTENSIONS
Offset corrected R. J. Mathar, Jan 26 2009
STATUS
approved