Thread Rating:
  • 1 Vote(s) - 3 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e))
#55
Hej Gotfried,

Gottfried Wrote:Hmm; I don't know absolutely. See my derivations in the article
function; it seems pretty straightforward and I don't think, there are other solutions - but...

t is t=exp(u); so even if we use periodical u (including multiple 2*Pi*I*k) there is only one t according to my considerations. Don't know, whether there are other solutions possible - but if you should see another solution, I'd like to know this.

Gottfried


Why not:

Define complex number u Euler style , as
u=a+sqrt(-1)b so u= a+-ib where +i and -i must be used separately but together- they are both in sqrt(-1) always. And stick to it throughout the derivations.

Then red point coordinates will be:

u= 0+-ipi/2
t=0+-i
s=e^pi/2

And that will correspond to h(e^pi/2) having values +i, -i in at least in first 2 branches. So on right quadrants will have 2 curves instead of 1. May be symmetric, may be not- it has to be calculated according to Your methods. On left quadrants I am not sure as I can not really understand what values in left quadrants correspond to. .

and then for complex-> real case
[b]0<beta<pi taken anticlockwise AND 0<beta<-pi taken clockwise
[/b]
Alfa, s limits i do not know-may be they change as well in this case?

Ivars
Reply


Messages In This Thread
RE: Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e)) - by Ivars - 11/14/2007, 10:10 PM
RE: Tetration below 1 - by Gottfried - 09/09/2007, 07:04 AM
RE: The Complex Lambert-W - by Gottfried - 09/09/2007, 04:54 PM
RE: The Complex Lambert-W - by andydude - 09/10/2007, 06:58 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
  b^b^x with base 0<b<e^-e have three real fixpoints Gottfried 1 1,968 11/07/2017, 11:06 AM
Last Post: sheldonison
  2 real fixpoints again ....... tommy1729 10 8,075 02/23/2016, 10:17 PM
Last Post: tommy1729
  A new set of numbers is necessary to extend tetration to real exponents. marraco 7 8,266 03/19/2015, 10:45 PM
Last Post: marraco
  Real-analytic tetration uniqueness criterion? mike3 25 19,224 06/15/2014, 10:17 PM
Last Post: tommy1729
  About real limits tommy1729 1 2,180 09/23/2013, 09:24 PM
Last Post: tommy1729
  Solutions to f ' (x) = f(f(x)) ? tommy1729 1 2,082 08/12/2013, 12:10 AM
Last Post: tommy1729
  Real and complex behaviour of the base change function (was: The "cheta" function) bo198214 39 47,182 08/13/2011, 06:33 PM
Last Post: bo198214
  The imaginary tetration unit? ssroot of -1 JmsNxn 2 5,033 07/15/2011, 05:12 PM
Last Post: JmsNxn
  Tetration and imaginary numbers. robo37 2 5,005 07/13/2011, 03:25 PM
Last Post: robo37
  Infinite tetration of the imaginary unit GFR 40 50,277 06/26/2011, 08:06 AM
Last Post: bo198214



Users browsing this thread: 1 Guest(s)