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The upper superexponential
#11
sheldonison Wrote:The "upper/lower" properties of these two sexp solutions are very interesting, especially being able to convert one to the other. The "upper" solution approaches the larger fixed point at -infinity, and the lower solution approaches the smaller fixed point at +infinity.

Can this be applied to Kneser's fixed point solution for bases larger than (e^(1/e))? For base e, Kneser's solution, has complex values at the real number line,

No, Kneser though starts with the Schröder function at the primary (non-real) fixed point, which gives a superlogarithm/superexponential that has non-real values on the real axis. The superexponential is entire. However his aim is to have a real analytic solution. So he applies a conformal map to make it real on the real the axis, paying with the entireness.

Quote: and the function approaches the fixed point as x grows towards +infinity.
Which function? The superexponential surely not. But superlogarithm/Abel function approaches the primary fixed point for .

Quote: But the desired solution has real values for all x>-2, and complex values for all x<-2 (except for the singularities). Moreover, the desired solution approaches the fixed point, as x approaches -infinity.
...
This has probably already been done, but can Kneser's base e solution, approaching a complex fixed point at +infinity, be converted it to another solution, approaching the fixed point at -infinity, with real values at the real number line, for all x>-2? Perhaps this line of reasoning isn't applicable because the solution wouldn't be holomorphic for all x>-2.

I think basically its already what Kneser did, however I dont know whether his solution approaches a fixed point for . This is also rather doubtful, because the development at the conjugate fixed point should bring the same solution. It can not converge to the non-real fixed point and to its conjugate at the same time.

Did you have a look at my introduction to Kneser's superlogarithm?
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#12
This post has two authors, Dmitrii Kouznetsov and Henryk Trappmann.
We consider the superexponentials on base .
In addition to the upper superexponential (which is always bigger than the upper fixed point along the real axis) and the lower superexponential (which has values less than the lower fixed point along the real axis),
we plot and discuss also the two "intermediate" superexponentials with values between 2 and 4.
We denote the superexponentials as . The first subscript indicates the fixed point, used to develop the corresponding Schroeder function. For base it is either 2 or 4. There exist *two* classes of non-trivial real-analytic regular super-exponentials at each fixed point: One with values above the fixed point and one with a values below the fixed point (if we count the tricial constant function, that is equal to the fixed point, then there are *three* real-analytic regular super-exponentials at each fixed point.) In order to distingish these functions, we use the second subscript. This subscript indicates the value of the super-exponential at zero.
In such a way, for each fixed point, there are two classes of super-functions. Each class contains all the functions that are translations along the x-axis of the given example function. Below, for each fixed point, ( and ), we plot one function of each of these two classes, id est, four functions.
Additionally to the already introduced super-exponentials and we present here the super-exponentials and . Which however are not distinguishable at the real axis:
   
For comparison, the tetrational tet to base as obtained with Dmitriis Cauchy-integral algorithm is also drawn.

There is no hope to distinguish and at the screen even at a crazy zoom-in. The difference is really small. It is smaller than a pixel at your screen. It is smaller than wavelength of light we use to see this picture. It is even smaller, than atoms, of which your computer consists...
In order to show that these two functions are not the same, we define the function , which is plotted with factor , id est, , in dark pink at the bottom.
And even after so strong scaling-up, the difference remains smaller than unity; it oscillates along the real axis, passing through zero at integer values of its argument and decaying at .

Each of functions satisfies the same equation , at least in vicinity of the origen of coordinates and in the positive direction of the real axis. These functions are periodic, for all that belong to the range of holomorphism. The periods are and , id est, pure imaginary.
We tried to make the range of holomorphism of these fuinctions so large as possible, in order to exclude the functions that can be obtained from function with modification of the argument: , where is 1-periodic function, holomorphic at least in some vicinity of the real axis, such that . The mofigied function satisfies the same equation; . However, while is not identically zero, the modification destroys the periodicity of function, allthough the function may be also analytic and even entire (for the case ). Therefore, we need to keep the periodicity as the criterion, necessary for the uniqueness of these functions. As the periods are imaginary, the extension of functions to the complex plane seems to be the only way to provide the uniqueness.

Functions above are related: can be expressed through and can be expressed through with some complex constant ofsets of the arguments.
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#13
(05/10/2009, 02:13 PM)Kouznetsov Wrote: ... we present here the super-exponentials and . Which however are not distinguishable at the real axis:
...
We tried to make the range of holomorphism of these fuinctions so large as possible, in order to exclude the functions that can be obtained from function with modification of the argument: , where is 1-periodic function, holomorphic at least in some vicinity of the real axis, such that . The mofigied function satisfies the same equation; . However, while is not identically zero, the modification destroys the periodicity of function, allthough the function may be also analytic and even entire (for the case ). Therefore, we need to keep the periodicity as the criterion, necessary for the uniqueness of these functions. As the periods are imaginary, the extension of functions to the complex plane seems to be the only way to provide the uniqueness.

Functions above are related: can be expressed through and can be expressed through with some complex constant ofsets of the arguments.
First of all, thanks for the post, this is very interesting material. Are and the same two functions in the Bummer post?

Could you comment on how the behavior of the two functions differ in the complex plane? Do both functions have the same values at z=+/-i*infinity? Do they have the same periodicity? Does only one have singularities?

Given that at all integer values of z, then can these two functions be expressed in terms of each other, where ? Is the function analytic?

- Sheldon
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#14
(05/11/2009, 12:55 PM)sheldonison Wrote: Are and the same two functions in the Bummer post?

yes. They are also related to the earlier post which considers the two regular half iterates of on the interval (2,4), these are:

and
.

Quote:Could you comment on how the behavior of the two functions differ in the complex plane?
... Do they have the same periodicity?
... Does only one have singularities?

is entire, has period .
is not entire, has period .
Dmitrii can perhaps tell more about the singularities.

Quote: Do both functions have the same values at z=+/-i*infinity?
They have no limit along the imaginary axis because they are imaginary periodic.

Quote:Given that at all integer values of z, then can these two functions be expressed in terms of each other, where ? Is the function analytic?

Yes, is analytic, though may somewhere have non-real singularities.
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#15
(05/11/2009, 01:21 PM)bo198214 Wrote: is entire, has period .
is not entire, has period .
....
They have no limit along the imaginary axis because they are imaginary periodic.
When I first looked at Dimitrii's graphs in "Bummer", I didn't realize that the two functions were completely different functions in the imaginary plane, and have different imaginary periods! What I noticed was one had cut points, and the other had fractal behavior. Are the imaginary periods exactly repeating copies?

(05/10/2009, 02:13 PM)Kouznetsov Wrote: Functions above are related: can be expressed through and can be expressed through with some complex constant ofsets of the arguments.

The fractal behavior of is increasing to infinity via tetration, except it is occurring at the i=imaginary_period/2 line, with real values! But otherwise, the fractal behavior is as one would expect! It sounds as though the conversions are as simple as:



,

Where the complex offset is just a real offset plus half of the imaginary period of each function.
This means along with the complex offsets, also allows conversions between and , the lower superexponential, and the upper superexponential.
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#16
(05/11/2009, 08:12 PM)sheldonison Wrote: Are the imaginary periods exactly repeating copies?

yes. .

Quote:The fractal behavior of is increasing to infinity via tetration, except it is occurring at the i=imaginary_period/2 line, with real values!

Yes! On the imaginary axis they are just translated by . Isnt that strange!

Quote:It sounds as though the conversions are as simple as:



,


Where the complex offset is just a real offset plus half of the imaginary period of each function.

Absolutely!

Quote:This means along with the complex offsets, also allows conversions between and , the lower superexponential, and the upper superexponential.

But can not computed directly.
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#17
(05/11/2009, 12:55 PM)sheldonison Wrote: Are and the same two functions in the Bummer post?
Yes. These are the same functions.
Henryk asked me to plot them all versus real argument as a separate post.

(05/11/2009, 12:55 PM)sheldonison Wrote: Could you comment on how the behavior of the two functions differ in the complex plane?
Yes.
Functions and entire.
One of them can be obtained from another one, just displacing the argument.

Tetration has, as you know, singularities and the cutline; due to the periodicity, there is set of singulatities and cutlines. Function can be obtained by translation of tetration , so, it has similar singulatities.
[/quote]

(05/11/2009, 12:55 PM)sheldonison Wrote: Do both functions have the same values at z=+/-i*infinity?
No. There is no need to talk about values ,
because each of them is periodic and the periods are imaginary.

(05/11/2009, 12:55 PM)sheldonison Wrote: Do they have the same periodicity?
No. Periods are different:



(05/11/2009, 12:55 PM)sheldonison Wrote: Does only one have singularities?
Tetration has singularities;
its displacement has too.

(05/11/2009, 12:55 PM)sheldonison Wrote: Given that at all integer values of z, then can these two functions be expressed in terms of each other, where ?
Yes.

(05/11/2009, 12:55 PM)sheldonison Wrote: Is the function analytic?
Yes.

is 1-periodic function; it is almost sinusoidal.

Henryk, can we begin to distribute the draft of our paper?
It would answer a lot of questions we provoked with the plot...
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#18
(05/12/2009, 08:54 AM)Kouznetsov Wrote: Henryk, can we begin to distribute the draft of our paper?
It would answer a lot of questions we provoked with the plot...

Yes. I posted it just in the thread Laplace transform of tetration.
Or you can directly open it here.
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#19
I've a small contribution to this thread, however while I initially thought I could provide a nice solution to give an eich-value for synchronizing the fixpoint-specific results and a functional relation between the values of the two fixpoint-specific versions, this idea finally failed, but maybe there is still some extentable idea in it.



We look at the curve for b^x , where b=sqrt(2), and find the three segments

s0 = -inf..(2
s1 = 2)..(4
s2 = 4).. inf

corresponding to the fixpoints.

We also look at the display of the same thing in terms of the function, depending on h, so

f(h) = exp_b°h(x)

where we assign to x some constant value and write f_2(h) for the function developed at fixpoint 2 and f_4(h) for that at fixpoint 4.

In the graphs for f(h) in the beginning of this thread
[Image: attachment.php?aid=491]
(see Dmitri's graph) we have a natural norm for the horizontal centering of the curve in the segment s0; we simply have, that at height h=0 the value of f(h) is x, or if as usual x=1, then f(0) = 1 - and this gives the required norm, or "natural center", for the curve.

For the curves in the other segments s1 and s2 we don't have such a natural center, it was just arbitrarily taken as h=0 where x=3 and f(0)=x for the segment s1 and x=5 f(0)=x for the segment s2.


Now we consider only the curve in the segment s1.

On the other hand we look at the differences of f(h) when developed at the different fixpoints t0=2 and t1=4 in the segment s1. A rough curve for f_2(h)-f_4(h) is also given in the plot of Dmitri; it is nearly sinusoidal with the height-parameter h and periodic with h (mod 1).

However, in some more detailed versions of our graphs we can see, that the curve of the differences not only has decreasing amplitude when h approaches the infinity, but also, in general is not very well symmetric wrt x-axis. ( see some curves in the plot some curves in the posting in the thread)

Here comes now my observation/idea.

The asymmetry depends on the initial choice of x; if we select a "good" x, the curve of differences is nicely symmetric wrt to the x-axis (or differently stated: to the occuring y-values). That also means, that the fixpoint-2 curve and fixpoint4-curve are sometimes in phase and sometimes not, depending on the "eich-value" x for h=0.

I was searching for some special property of such x. and the striking finding was, that if I used the imaginary height of Pi*I, beginning at x=1, then the curve of differences is nicely symmetric and "in phase".

Practically it is simply to negate the value of the schröder-function at x=1.
Example:
If we use the fixpoint-2-function, compute
Code:
´x0=1
a0 = schr(x0/t0-1)
a1 = - a0
x1 = t0*(1+schr°-1(a1))

we get some (real!) value x1 in the segment s1, which taken as norm for that segment x=x1, f(0)=x1 gives a very accurate symmetric image for the curve of differences.

However - I first thought here I got the perfect match, but this was only a near-match.
Instead I needed a binary search to find an even better norm-value, where symmetry was perfect; this could be seen, when not only the integer iterates but also the half-iterates f_2(0.5) and f_4(0.5) in that segment were equal. Then also the quarter-iterates f_2(0.25) - f_4(0.25) give the maximal difference dmax and using that value for the definition of the amplitude of a sinus-function amp=dmax one can relate the two functions by

Code:
´find optimal eich-value for x, set this for h=0
then
      amp = f_4(0.25)-f_2(0.25)

and the functional relation

     f_2(h) = f_4(h+ amp*sin(h*2*Pi))


This is a very nice functional relation.

...

This *were* a very nice functional relation...
However - again this is not the end. There is again a (smaller though) difference of the curves, and again not perfectly in phase.
May be one can continue this using fourier-analysis to some reasonable extend; unfortunately I've no clue about this matter.

So - my idea was to propose one eich-value x for the segments; I didn't actually succeed but the consideration may not be without worth. I think, the idea to use the value at the height I*Pi iterated from 1 has some charme, maybe someone finds an idea how to realize that.

Gottfried
Gottfried Helms, Kassel
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