A135544 Decimal expansion of (-1)^(I Pi).

0, 0, 0, 0, 5, 1, 7, 2, 3, 1, 8, 6, 2, 0, 3, 8, 1, 2, 3, 0, 6, 1, 4, 5, 4, 6, 5, 0, 9, 0, 3, 8, 2, 3, 9, 3, 6, 9, 5, 5, 7, 8, 7, 6, 9, 6, 9, 8, 3, 6, 6, 8, 0, 8, 9, 4, 1, 4, 2, 7, 6, 5, 8, 8, 1, 8, 4, 7, 1, 6, 8, 3, 1, 5, 1, 0, 3, 2, 3, 0, 5, 6, 7, 6, 2, 0, 6, 8, 5, 5, 9, 8, 1, 9, 5, 3, 1, 9, 3, 3, 3

Offset: 0

Author

Marvin Ray Burns, Feb 22 2008, Feb 23 2008

Keywords

• nonn
• cons

Examples

			(-1)^(I*Pi) = exp(-Pi)^(Pi) = 0.000051723186...
		

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.

Crossrefs

Cf. A057521, A002388, A093580.

Programs

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 *)

PARI

exp(-Pi^2) \\ Charles R Greathouse IV, Jan 23 2025

PARI

real((-1)^(I*Pi)) \\ Charles R Greathouse IV, Jan 23 2025

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].

Extensions

Offset corrected R. J. Mathar, Jan 26 2009