cp's OEIS Frontend

A272875 Decimal expansion of the real part of the infinite nested power (1+(1+(1+...)^i)^i)^i, with i being the imaginary unit.

%I A272875 #13 May 26 2023 05:42:39
%S A272875 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 A272875 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 A272875 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 A272875 Decimal expansion of the real part of the infinite nested power (1+(1+(1+...)^i)^i)^i, with i being the imaginary unit.
%C A272875 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 A272875 The imaginary part and the modulus of this complex constant are in A272876 and A272877, respectively.
%H A272875 Stanislav Sykora, <a href="/A272875/b272875.txt">Table of n, a(n) for n = 0..2000</a>
%F A272875 z0 = a+A272876*i satisfies the equations (1+z0)^i = z0, (1+z0)*z0^i = 1.
%e A272875 0.6738813311078755157802311904681019338764503347933725454899813516...
%t A272875 RealDigits[Re[z /. FindRoot[(1 + z)^I == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* _Amiram Eldar_, May 26 2023 *)
%o A272875 (PARI) \\ f(x) computes (x+(x+...)^i)^i, provided that it converges:
%o A272875 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 A272875 \\ To compute this constant, use:
%o A272875 z0 = f(1); real(z0)
%Y A272875 Cf. A156548, A272876, A272877.
%K A272875 nonn,cons
%O A272875 0,1
%A A272875 _Stanislav Sykora_, May 15 2016