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%I #13 May 26 2023 05:42:39
%S 6,7,3,8,8,1,3,3,1,1,0,7,8,7,5,5,1,5,7,8,0,2,3,1,1,9,0,4,6,8,1,0,1,9,
%T 3,3,8,7,6,4,5,0,3,3,4,7,9,3,3,7,2,5,4,5,4,8,9,9,8,1,3,5,1,6,6,9,3,1,
%U 2,1,3,0,3,2,7,5,2,9,3,8,0,7,5,0,8,5,0,7,6,3,1,1,7,3,1,7,9,4,1,6,3,6,1,0,1
%N Decimal expansion of the real part of the infinite nested power (1+(1+(1+...)^i)^i)^i, with i being the imaginary unit.
%C The mapping M(z)=(1+z)^i has in C a unique invariant point, namely z0 = a+A272876*i, which is also its attractor. Iterative applications of M applied to any starting complex point z (except for the singular value -1+0*i) rapidly converge to z0. The convergence, and the existence of this limit, justify the expression used in the name. It is easy to show that, close to z0, the convergence is exponential, with the error decreasing approximately by a factor of abs(z0/(1+z0))=0.4571... per iteration.
%C The imaginary part and the modulus of this complex constant are in A272876 and A272877, respectively.
%H Stanislav Sykora, <a href="/A272875/b272875.txt">Table of n, a(n) for n = 0..2000</a>
%F z0 = a+A272876*i satisfies the equations (1+z0)^i = z0, (1+z0)*z0^i = 1.
%e 0.6738813311078755157802311904681019338764503347933725454899813516...
%t RealDigits[Re[z /. FindRoot[(1 + z)^I == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* _Amiram Eldar_, May 26 2023 *)
%o (PARI) \\ f(x) computes (x+(x+...)^i)^i, provided that it converges:
%o f(x)={my(z=1.0,zlast=0.0,eps=10.0^(1-default(realprecision)));while(abs(z-zlast)>eps,zlast=z;z=(x+z)^I);return(z)}
%o \\ To compute this constant, use:
%o z0 = f(1); real(z0)
%Y Cf. A156548, A272876, A272877.
%K nonn,cons
%O 0,1
%A _Stanislav Sykora_, May 15 2016
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