

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
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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 Wfunction. 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]] (* JeanFrançois 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=(zlog(z))/(11/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



