A059526 Decimal expansion of real part of solution to z = log z.

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

Offset: 0

Author

Fabian Rothelius, Jan 21 2001

Keywords

• cons
• nonn
• nice

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

Examples

			z = 0.31813150520476413531265425158766451720351761387139986692237... + 1.33723570143068940890116214319371061253950213846051241887631... *i
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..2000
• Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.
• Wolfram Research, FixedPoint

Crossrefs

Imaginary part is A059527.

Cf. A030178.

Cf: A277681 (another fixed point of exp(z)).

Programs

Mathematica

RealDigits[ Re[ N[ FixedPoint[ Log, 1 + I, 910], 105]]] [[1]]
RealDigits[ N[ Re[ ProductLog[-1]], 105]][[1]] (* Jean-François Alcover, Feb 01 2012 *)
RealDigits[Re[x/.FindRoot[x-Log[x]==0,{x,.5,1},WorkingPrecision->200]],10,120][[1]] (* Harvey P. Dale, Aug 07 2022 *)

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

Extensions

More terms from Vladeta Jovovic, Feb 26 2001

Edited and extended by Robert G. Wilson v, Aug 22 2002