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%I #20 May 30 2023 02:20:15
%S 2,6,9,2,9,3,4,3,7,1,6,9,3,1,1,2,2,7,1,9,0,8,6,8,0,2,1,2,6,8,8,6,2,0,
%T 1,0,5,3,2,9,1,1,0,0,6,0,3,7,6,8,4,6,7,1,7,1,2,7,1,6,0,1,5,1,5,2,8,3,
%U 9,2,3,1,5,2,6,4,9,8,1,7,6,1,9,8,3,1,3,6,8,0,1,9,9,1,0,9,8,9,9,9,4,8,8,4,1
%N Decimal expansion of the real part of the solution of z = (i+z)^i in C (i is the imaginary unit).
%C In C, the unique invariant point of the mapping M(z) = (i+z)^i is also its attractor. The convergence is linear and takes about 1650 iterations to reduce the value of |z - M(z)| by 1000 decimal digits. The imaginary part of the invariant point is in A290410.
%C Considering the definition, one can symbolically write A290409 + i*A290410 = (i+(i+(i+...)^i)^i)^i.
%H Stanislav Sykora, <a href="/A290409/b290409.txt">Table of n, a(n) for n = 0..2000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ComplexExponentiation.html">Complex Exponentiation</a>.
%e 0.269293437169311227190868021268862010532911006037684671712716015...
%t RealDigits[Re[z /. FindRoot[(I + z)^I == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* _Amiram Eldar_, May 30 2023 *)
%o (PARI) \p 3000 \\ Set precision
%o M(z)=(z+I)^I; \\ Mapping M
%o z=1.0;for(k=1,2000,z=M(z)); \\ Initialize and iterate
%o d = -floor(log(abs(z-M(z)))/log(10)) \\ A crude convergence test
%o real(z) \\ The result; keep << d digits, and test for stability.
%Y Cf. A272875, A272876, A272877, A290408, A290410.
%K nonn,cons
%O 0,1
%A _Stanislav Sykora_, Jul 30 2017
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