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 A059526 Decimal expansion of real part of solution to z = log z. 9
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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. 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 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 REFERENCES Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, http://www.ebyte.it/library/docs/math16/2016_MATH_Sykora_FixedPointsExp.pdf; DOI: 10.3247/SL6Math16.002, 2016. Wolfram Research, Mathematica, Version 4.1.0.0, Help Browser, under the function FixedPoint. LINKS Stanislav Sykora, Table of n, a(n) for n = 0..2000 Wolfram Research, FixedPoint EXAMPLE z = 0.31813150520476413531265425158766451720351761387139986692237... + 1.33723570143068940890116214319371061253950213846051241887631... *i MATHEMATICA RealDigits[ Re[ N[ FixedPoint[ Log, 1 + I, 910], 105]]] [[1]] RealDigits[ N[ Re[ ProductLog[-1]], 105]][[1]] (* Jean-François Alcover, Feb 01 2012 *) PROG (PARI) z=I; for(k=1, 16000, z=log(z)); real(z) \\ Stanislav Sykora, Jun 07 2015 \\ Using realprecision \p 2010 (PARI) z=I; for(k=1, 10, z-=(z-log(z))/(1-1/z)); real(z) \\ Jeremy Tan, Sep 23 2017 CROSSREFS Imaginary part is A059527. Cf. A030178. Cf: A277681 (another fixed point of exp(z)). Sequence in context: A165498 A195731 A154294 * A091839 A155789 A179393 Adjacent sequences:  A059523 A059524 A059525 * A059527 A059528 A059529 KEYWORD cons,nonn,nice AUTHOR Fabian Rothelius, Jan 21 2001 EXTENSIONS More terms from Vladeta Jovovic, Feb 26 2001 Edited and extended by Robert G. Wilson v, Aug 22 2002 STATUS approved

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Last modified October 16 09:29 EDT 2019. Contains 328056 sequences. (Running on oeis4.)