Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Universal uniqueness criterion II
#1
Lets summarize what we have so far:

Proposition. Let be a vertical strip somewhat wider than , i.e. for some and .
Let for some and let be two domains (open and connected) for values, and let be holomorphic on . Then there is at most one function that satisifies
(1) is holomorphic on and
(2) is real and strictly increasing on
(3) for all and
(4) There exists an inverse holomorphic function on , i.e. a holomorphic function such that for all .

Proof. Let be two function that satisfy the above conditions. Then the function is holomorphic on (because and (4)) and satisfies . By (3) and (4)
and .
So can be continued from to an entire function and is real and strictly increasing on the real axis.

by our previous considerations.
By Big Picard every real value of is taken on infinitely often if is not a polynomial, but every real value is only taken on once on the real axis, thatswhy still . But this is in contradiction to . So must be a polynomial that takes on every real value at most once. This is only possible for with because ..

In the case of tetration one surely would chose and or . However I am not sure about the domain which must contain and hence give some bijection , with some .

Of course in the simplest case one just chooses if one has some function in mind already. However then we can have a different function with but our intention was to have a criterion that singles out other solutions.
So we need an area on which every slog should be defined at least and satisfy as well as .
Reply
#2
I just see that we can essentially simplify our conditions:

Proposition. Let be a vertical strip somewhat wider than , i.e. for some and .
Let , , be three domains (open and connected) such that , and let be holomorphic on , let . Then there exist at most one function that satisifies
(1) is holomorphic on and
(2) for all and
(3) There is a such that is biholomorphic.

Proof. Let be two function that satisfy the above conditions. Then the function is holomorphic on (because and (3)) and satisfies . By (3) and (4)
and .
So can be continued from to an entire function.
But the same is also true for by the same reasoning. But as and , we see that is a bijection!
Again with Picard's big theorem we conclude that with .
Reply
#3
bo198214 Wrote:...Let for some ...
Henryk: For tetration, I would define
for some ...
then I like your proof.

I include below the small part of
http://math.eretrandre.org/tetrationforu...77#pid2477
which is picture of slog(S), assuming, and that is small and not seen. Vertical lines correspond to
and

Horisontal lines correspond to
and

The curvilinear mesh is produced by images of lines
and
.
The pink cutline corresponds to .
The images of lines with integer are a little bit extended.
Also, images of lines and are shown.
There is biholomorphizm .
   

Henrik,
1. Do you plan to polish this proof or I may include this into the paper?
2. Can we claim, that some of singularities of a modified tetration are at
? (In this case, we can include the case at once).
Reply
#4
Kouznetsov Wrote:
bo198214 Wrote:...Let for some ...
Henryk: For tetration, I would define
for some ...
then I like your proof.

Yes that was a mistype, it should read . I change that in the original post.

Quote:I include below the small part of
http://math.eretrandre.org/tetrationforu...77#pid2477
which is picture of slog(S),

thanks, for the illustration.

Quote:1. Do you plan to polish this proof or I may include this into the paper?
Its not yet finished, we still have no universal domain for the slog, which we need to have uniqueness independent of the specific domain .

Quote:2. Can we claim, that some of singularities of a modified tetration are at
?

Dmitrii, modified tetration is not all, if you want to consider then you need to have a first, a that is holomorphic on which in turn imposes conditions on the cut, set in the domain of definition of .
Reply
#5
bo198214 Wrote:
Kouznetsov Wrote:2. Can we claim, that some of singularities of a modified tetration are at
?
Dmitrii, modified tetration is not all, if you want to consider then you need to have a first, a that is holomorphic on which in turn imposes conditions on the cut, set in the domain of definition of .
Sorry, I try again: If is entire and 1-periodic and , then can we claim that ?
Reply
#6
Kouznetsov Wrote:I try again: If is entire and 1-periodic and , then can we claim that ?

Hm, sorry, I dont know. Wherefore do you need such a statement?
Reply
#7
bo198214 Wrote:
Kouznetsov Wrote:I try again: If is entire and 1-periodic and , then can we claim that ?
Hm, sorry, I dont know..
ok, I do know. Conjectiure:
Let
be entire 1-periodic non constant function,
,
for all in some vicinity of the real axis
for all complex
Then there exist such that and

bo198214 Wrote:Wherefore do you need such a statement?
Yes, I do. Then we have beautiful and general proof of uniqueness of tetration as it is defined at http://en.citizendium.org/wiki/Tetration

P.S. For other participants, I repeat here the essense from one of our previous discussions.
Many times I tried to build-up an example to negate the conjecture above.
Therefore I claim this conjecture.
I agree with you, that it is not sufficient reason for such a claim.
(Although I never met a simple condition for an existing function, such that I could not provide an example.)
It would be interesting to construct the sufficient reason. (then the conjecture becomes Theorem)
Reply
#8
bo198214 Wrote:Proof. Let be two function that satisfy the above conditions. Then the function is holomorphic on (because and (3)) and satisfies ...
Why ? There was no above. Should not be ?
Reply
#9
Kouznetsov Wrote:
bo198214 Wrote:Proof. Let be two function that satisfy the above conditions. Then the function is holomorphic on (because and (3)) and satisfies ...
Why ? There was no above. Should not be ?

and are two functions that satisfy the above conditions, as I wrote. Perhaps write better: "Let and be two solutions of the above conditions."
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Semi-exp and the geometric derivative. A criterion. tommy1729 0 1,053 09/19/2017, 09:45 PM
Last Post: tommy1729
  A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov 13 9,396 02/17/2017, 05:21 AM
Last Post: JmsNxn
  Uniqueness of half-iterate of exp(x) ? tommy1729 14 12,736 01/09/2017, 02:41 AM
Last Post: Gottfried
  Removing the branch points in the base: a uniqueness condition? fivexthethird 0 1,311 03/19/2016, 10:44 AM
Last Post: fivexthethird
  [2014] Uniqueness of periodic superfunction tommy1729 0 1,694 11/09/2014, 10:20 PM
Last Post: tommy1729
  Real-analytic tetration uniqueness criterion? mike3 25 18,676 06/15/2014, 10:17 PM
Last Post: tommy1729
  exp^[1/2](x) uniqueness from 2sinh ? tommy1729 1 1,954 06/03/2014, 09:58 PM
Last Post: tommy1729
  Uniqueness Criterion for Tetration jaydfox 9 10,170 05/01/2014, 10:21 PM
Last Post: tommy1729
  Uniqueness of Ansus' extended sum superfunction bo198214 4 6,048 10/25/2013, 11:27 PM
Last Post: tommy1729
  A question concerning uniqueness JmsNxn 3 5,037 10/06/2011, 04:32 AM
Last Post: sheldonison



Users browsing this thread: 1 Guest(s)