A059527 Decimal expansion of imaginary part of solution to z = log z.

1, 3, 3, 7, 2, 3, 5, 7, 0, 1, 4, 3, 0, 6, 8, 9, 4, 0, 8, 9, 0, 1, 1, 6, 2, 1, 4, 3, 1, 9, 3, 7, 1, 0, 6, 1, 2, 5, 3, 9, 5, 0, 2, 1, 3, 8, 4, 6, 0, 5, 1, 2, 4, 1, 8, 8, 7, 6, 3, 1, 2, 7, 8, 1, 9, 1, 4, 3, 5, 0, 5, 3, 1, 3, 6, 1, 2, 0, 4, 9, 8, 8, 4, 1, 8, 8, 8, 1, 3, 2, 3, 4, 3, 8, 7, 9, 4, 0, 1, 5, 6, 1, 0, 3, 8

Offset: 1

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.

Examples

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

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000
• Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.
• Wolfram Research, FixedPoint

Crossrefs

Real part is A059526.

Cf. A030178.

Programs

Mathematica

RealDigits[ Im[ N[ FixedPoint[ Log, 1 + I, 910], 105]]] [[1]]
RealDigits[ N[ Im[ ProductLog[-1]], 105]][[1]] (* Jean-François Alcover, Feb 01 2012 *)

PARI

z=I;for(k=1,16000,z=log(z));imag(z)  \\ Using realprecision \p 2010. - Stanislav Sykora, Jun 07 2015

PARI

z=I; for(k=1, 10, z-=(z-log(z))/(1-1/z)); imag(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