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A202501 Decimal expansion of x satisfying x=e^(-Pi*x/2).

%I A202501 #26 Feb 07 2025 16:44:07
%S A202501 4,7,4,5,4,0,9,9,9,5,1,2,6,5,1,1,2,3,0,1,7,4,6,7,9,4,4,0,4,8,2,1,2,4,
%T A202501 5,1,1,4,9,1,0,7,6,8,0,6,5,9,9,2,6,7,1,4,0,9,8,1,3,7,9,7,2,2,7,0,6,8,
%U A202501 8,5,5,9,8,9,9,3,3,0,8,8,5,9,8,3,1,1,4,9,3,2,0,7,0,0,5,9,0,5,9
%N A202501 Decimal expansion of x satisfying x=e^(-Pi*x/2).
%C A202501 See A202348 for a guide to related sequences. The Mathematica program includes a graph.
%C A202501 Also the only solution of x=I^(x*I), since I^I = exp(-Pi/2). Also the infinite power tower (tetration) of I^I, i.e., the convergent sequence I^(I*I^(I*I^(...(I*I^I)...))). Also LambertW(Pi/2)/(Pi/2). - _Stanislav Sykora_, Nov 06 2013
%H A202501 Stanislav Sykora, <a href="/A202501/b202501.txt">Table of n, a(n) for n = 0..2000</a>
%H A202501 Steven R. Finch, <a href="/A202501/a202501.pdf">Tauberian Constants</a>, August 30, 2004 [Cached copy, with permission of the author]
%e A202501 x=0.474540999512651123017467944048212451149107680...
%t A202501 u = -Pi/2; v = 0;
%t A202501 f[x_] := x; g[x_] := E^(u*x + v)
%t A202501 Plot[{f[x], g[x]}, {x, -1, 1}, {AxesOrigin -> {0, 0}}]
%t A202501 r = x /. FindRoot[f[x] == g[x], {x, .4, .5}, WorkingPrecision -> 110]
%t A202501 RealDigits[r]    (* A202501 *)
%t A202501 RealDigits[ 2*ProductLog[Pi/2]/Pi, 10, 99] // First (* _Jean-François Alcover_, Feb 27 2013 *)
%o A202501 (PARI) lambertw(Pi/2)/(Pi/2) \\ _Stanislav Sykora_, Nov 06 2013
%Y A202501 Cf. A202348, A049006, A231095 (comment).
%K A202501 nonn,cons
%O A202501 0,1
%A A202501 _Clark Kimberling_, Dec 20 2011