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))
Ivars Wrote:Hej Gotfried,

Congratulations! There are no accidents.

Quote:So how do you finally compute the th branch of or with this formula?

Yes, how would You? And particularly, in case

h(e^pi/2) and h(e^-pi/2)?

Waiting impatiently evenSmile


No chance.... ;-)

Yes, there is the coincidence; but I did not compute the W-function but the function, which gives real bases b (or, to avoid confusion with the parameter-notation in the article:bases s) for exp(u/t) where s=t^(1/t) is the principal branch, given u (or more precisely: given the imaginary part of u), and get real bases s>1 .

I set imag(u) = beta = any real value -pi < beta < pi
compute real(u) according to the above formula thus having u completed,
then compute t = exp(u)
and then compute s = exp(u/t), which is then surely real.

The formula for this is in the file fixpoints.pdf
In this article I've not yet included the extension of the range for beta>pi; but I've done some computations with this and found further fixpoints for the real s in the regions 2*k*pi< beta < 2*(k+1)*pi. However, there is no exact periodicity, the consecutive fixpoints for the same s approach the lower bound 2*k*pi with the index k.

I can find the inverse, to compute a fixpoint t by a given s, only by numerical approximation, since s=f(u) is monotonic, using a binary iterative process.
If I *had* the branch-enabled Lambert-W-function, this would be easier, but Pari/GP doesn't have it and I still have only the python-example from wikipedia for the real-valued region, and have not yet invested in programming it myself.

Gottfried Helms, Kassel

Messages In This Thread
RE: Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e)) - by Gottfried - 11/07/2007, 12:58 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
  Constructing real tetration solutions Daniel 4 4,301 12/24/2019, 12:10 AM
Last Post: sheldonison
  b^b^x with base 0<b<e^-e have three real fixpoints Gottfried 1 4,098 11/07/2017, 11:06 AM
Last Post: sheldonison
  2 real fixpoints again ....... tommy1729 10 15,667 02/23/2016, 10:17 PM
Last Post: tommy1729
  A new set of numbers is necessary to extend tetration to real exponents. marraco 7 14,464 03/19/2015, 10:45 PM
Last Post: marraco
  Real-analytic tetration uniqueness criterion? mike3 25 33,547 06/15/2014, 10:17 PM
Last Post: tommy1729
  About real limits tommy1729 1 3,854 09/23/2013, 09:24 PM
Last Post: tommy1729
  Solutions to f ' (x) = f(f(x)) ? tommy1729 1 3,791 08/12/2013, 12:10 AM
Last Post: tommy1729
  Real and complex behaviour of the base change function (was: The "cheta" function) bo198214 39 72,912 08/13/2011, 06:33 PM
Last Post: bo198214
  The imaginary tetration unit? ssroot of -1 JmsNxn 2 7,588 07/15/2011, 05:12 PM
Last Post: JmsNxn
  Tetration and imaginary numbers. robo37 2 7,692 07/13/2011, 03:25 PM
Last Post: robo37

Users browsing this thread: 1 Guest(s)