Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Has anyone solved iterations of z+Γ(z)?
#1
Just out of curiosity. Has anyone solved the complex continuous iterations of the function ?
This will be very challenging because has no zeros, and the zero in the directed complex infinity is non-Botcher-constructable.

My thought is using a function to map the fixed point at to 0 with a specific function, may be the inverse of .
Reply
#2
A wild guess without basis, maybe this function is just a special case of Fox H Big Grin
Reply
#3
I see no reasons why iterations of this type should not be possible with the same techniques we used for tetration and related ones.
I once started a thread here about the superfunction of exp(z)+z.
Notice exp(z)+z also has no FINITE fixpoint just like your z + Γ(z).
But exp(z) + z has a fixpoint at negative infinity.

On the other hand , exp(z) + z has no poles.
And exp(z) + z has a nice asymp to z.
So the situation is different.

One could also wonder about the generalization z +  Γ(z,v) for various v.
In particular v = 1 for obvious reasons.
On the other hand Γ(z,1) does have zero's.

----

the infinite composition/functional equation :

f(s+1) = t(s) f(s) + Γ( t(s) f(s) )

- where t(s) is like with the gaussian method - should work , not ?


Interesting question, thank you.


regards

tommy1729
Reply
#4
Again , nice question !
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  [MSE-SOLVED] Subfunction is functorial!!!! MphLee 14 3,208 06/06/2021, 11:16 PM
Last Post: JmsNxn
  [MSE] Shape of orbit of iterations with base b on Shell-Thron-region Gottfried 14 18,014 12/13/2019, 02:33 PM
Last Post: Ember Edison
  Math overflow question on fractional exponential iterations sheldonison 4 9,165 04/01/2018, 03:09 AM
Last Post: JmsNxn
  Rational operators (a {t} b); a,b > e solved JmsNxn 30 67,849 09/02/2016, 02:11 AM
Last Post: tommy1729
  Periodic analytic iterations by Riemann mapping tommy1729 1 4,520 03/05/2016, 10:07 PM
Last Post: tommy1729
  Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x Ivars 10 22,855 03/29/2015, 08:02 PM
Last Post: tommy1729
  Zeta iterations Balarka Sen 11 20,246 02/26/2013, 09:49 AM
Last Post: Gottfried
  Means and intermediate operations (was: Rational operators (a {t} b); a,b > e solved) Cherrina_Pixie 3 8,979 06/14/2011, 09:52 PM
Last Post: JmsNxn
  2 real fixpoints ... 3 iterations ? tommy1729 4 12,162 08/10/2009, 11:45 PM
Last Post: bo198214
  Functional super-iteration and hierarchy of functional hyper-iterations Base-Acid Tetration 24 47,289 05/12/2009, 07:11 AM
Last Post: bo198214



Users browsing this thread: 1 Guest(s)