cp's OEIS Frontend

A164003 Decimal expansion of exp(-Pi^2/2).

%I A164003 #23 Jul 09 2025 04:12:44
%S A164003 0,0,7,1,9,1,8,8,3,3,5,5,8,2,6,3,6,5,6,0,7,8,0,1,3,6,6,3,9,6,3,7,1,2,
%T A164003 0,2,9,5,5,3,6,2,3,1,8,0,8,1,5,9,7,9,4,7,5,5,8,0,3,7,1,8,1,2,4,1,2,1,
%U A164003 3,1,8,8,6,9,7,5,6,6,5,0,8,9,5,9,6,1,0,7,9,4,3,9,8,6,5,6,6,2,3,3,0
%N A164003 Decimal expansion of exp(-Pi^2/2).
%C A164003 Arises in connection with the expansion (e i)^(Pi i) = e^(Pi i) * i^(Pi i) = -1 * exp(Pi i log i) = -exp(Pi i * Pi/2 i) = -exp(-Pi^2/2) = -0.00719188335582636560780136639...
%C A164003 One has to be careful about branches of multivalued complex functions. By definition (e i)^(Pi i) is exp(Pi i log(e i)) [using any of the branches of log] = exp(Pi i (1 + Pi i/2 + 2 n Pi i)) [ for any integer n ] = - exp(- Pi^2/2 - 2 n Pi^2). There is no imaginary part in any of its branches. If n=0 we get (-1) times the present constant.
%C A164003 Based on postings to the Sequence Fans Mailing List by _Alonso del Arte_, _Charles R Greathouse IV_, _Sean A. Irvine_, _Robert Israel_, and other correspondents
%H A164003 G. C. Greubel, <a href="/A164003/b164003.txt">Table of n, a(n) for n = 0..5000</a>
%e A164003 .0071918833558263656078013663963712029553623180815979475580371...
%t A164003 Join[{0,0},RealDigits[Exp[-Pi^2/2],10,120][[1]]] (* _Harvey P. Dale_, Jul 11 2011 *)
%o A164003 (PARI) exp(-Pi^2/2) \\ _Charles R Greathouse IV_, Mar 25 2014
%Y A164003 Cf. A135544.
%K A164003 nonn,cons
%O A164003 0,3
%A A164003 _N. J. A. Sloane_, Aug 06 2010