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%I #50 Aug 07 2022 16:19:11
%S 3,1,8,1,3,1,5,0,5,2,0,4,7,6,4,1,3,5,3,1,2,6,5,4,2,5,1,5,8,7,6,6,4,5,
%T 1,7,2,0,3,5,1,7,6,1,3,8,7,1,3,9,9,8,6,6,9,2,2,3,7,8,6,0,6,2,2,9,4,1,
%U 3,8,7,1,5,5,7,6,2,6,9,7,9,2,3,2,4,8,6,3,8,4,8,9,8,6,3,6,1,6,3,8,4,4,2,1,4
%N Decimal expansion of real part of solution to z = log z.
%C Repeatedly take logs, starting from any number not equal to 0, 1, e, e^e, e^(e^e), etc. and you will converge to 0.31813150... + 1.33723570...*I.
%C A complex number w with a negative imaginary part will converge to the conjugate of z since log(conjugate(w)) = conjugate(log(w)). - _Gerald McGarvey_, Mar 02 2009
%C This z and its conjugate are the only two complex solutions of z=log(z) on the principal branch of log(z), and of exp(z)=z for |arg(z)| <= Pi. They are also the only nontrivial (z!=0) principal branch solutions of z=W(z^2), W being the Lambert W-function. Though the two values are iterative attractors of the mapping z->log(z), the convergence is rather slow; the precision improves by slightly more than one binary bit every 2.25 iterations (about 7500 iterations are needed to make stable the first 1000 decimal digits). - _Stanislav Sykora_, Jun 07 2015
%H Stanislav Sykora, <a href="/A059526/b059526.txt">Table of n, a(n) for n = 0..2000</a>
%H Stanislav Sykora, <a href="https://doi.org/10.3247/SL6Math16.002">Fixed points of the mappings exp(z) and -exp(z) in C</a>, 2016.
%H Wolfram Research, <a href="https://reference.wolfram.com/language/ref/FixedPoint.html">FixedPoint</a>
%e z = 0.31813150520476413531265425158766451720351761387139986692237... + 1.33723570143068940890116214319371061253950213846051241887631... *i
%t RealDigits[ Re[ N[ FixedPoint[ Log, 1 + I, 910], 105]]] [[1]]
%t RealDigits[ N[ Re[ ProductLog[-1]], 105]][[1]] (* _Jean-François Alcover_, Feb 01 2012 *)
%t RealDigits[Re[x/.FindRoot[x-Log[x]==0,{x,.5,1},WorkingPrecision->200]],10,120][[1]] (* _Harvey P. Dale_, Aug 07 2022 *)
%o (PARI) z=I;for(k=1,16000,z=log(z));real(z) \\ _Stanislav Sykora_, Jun 07 2015 \\ Using realprecision \p 2010
%o (PARI) z=I; for(k=1, 10, z-=(z-log(z))/(1-1/z)); real(z) \\ _Jeremy Tan_, Sep 23 2017
%Y Imaginary part is A059527.
%Y Cf. A030178.
%Y Cf: A277681 (another fixed point of exp(z)).
%K cons,nonn,nice
%O 0,1
%A _Fabian Rothelius_, Jan 21 2001
%E More terms from _Vladeta Jovovic_, Feb 26 2001
%E Edited and extended by _Robert G. Wilson v_, Aug 22 2002
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