The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A272875 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A272875

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

#13 by Michel Marcus at Fri May 26 05:42:39 EDT 2023

STATUS

reviewed

approved

#12 by Joerg Arndt at Fri May 26 04:16:08 EDT 2023

STATUS

proposed

reviewed

#11 by Amiram Eldar at Fri May 26 03:42:57 EDT 2023

STATUS

editing

proposed

#10 by Amiram Eldar at Fri May 26 03:35:11 EDT 2023

MATHEMATICA

RealDigits[Re[z /. FindRoot[(1 + z)^I == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, May 26 2023 *)

STATUS

approved

editing

#9 by N. J. A. Sloane at Sat May 21 23:24:03 EDT 2016

STATUS

editing

approved

#8 by N. J. A. Sloane at Sat May 21 23:23:51 EDT 2016

COMMENTS

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 fast to z0. The convergence, and the existence of this limit, justify the expression used in the Namename. It is easy to show that, close to z0, the convergence is exponential, with the error decreasing approximately by the a factor of abs(z0/(1+z0))=0.4571... per iteration.

STATUS

proposed

editing

Discussion

Sat May 21

23:24

N. J. A. Sloane: a few small edits

#7 by Stanislav Sykora at Sun May 15 19:20:33 EDT 2016

STATUS

editing

proposed

#6 by Stanislav Sykora at Sun May 15 19:20:24 EDT 2016

COMMENTS

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 to any starting complex point z (except for the singular value -1+0*i) converge fast 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 the factor of abs(z0/(1+z0))=0.4571... per iteration.

STATUS

proposed

editing

#5 by Stanislav Sykora at Sun May 15 19:03:44 EDT 2016

STATUS

editing

proposed

#4 by Stanislav Sykora at Sun May 15 18:53:27 EDT 2016

COMMENTS

The mapping M(z)=(1+z)^i has in C a unique invariant point, namely z0 = a+A272876*i, which is also its attractor. Itarative Iterative applications of M to any starting complex point z, (except for the singular value -1+0*i, ) converge fast to z0. The convergence, and the existence of this limit justifies , justify the expression used in the Name.