05/10/2009, 02:13 PM
(This post was last modified: 05/11/2009, 12:52 AM by Kouznetsov.)
This post has two authors, Dmitrii Kouznetsov and Henryk Trappmann.
We consider the superexponentials on base \( b=\sqrt{2} \).
In addition to the upper superexponential (which is always bigger than the upper fixed point \( L=4 \) along the real axis) and the lower superexponential (which has values less than the lower fixed point \( L=2 \) along the real axis),
we plot and discuss also the two "intermediate" superexponentials with values between 2 and 4.
We denote the superexponentials as \( F_{L,d} \). The first subscript indicates the fixed point, used to develop the corresponding Schroeder function. For base \( \sqrt{2} \) 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, (\( L=2 \) and \( L=4 \)), we plot one function of each of these two classes, id est, four functions.
Additionally to the already introduced super-exponentials \( F_{2,1} \) and \( F_{4,5} \) we present here the super-exponentials \( F_{2,3} \) and \( F_{4,3} \). Which however are not distinguishable at the real axis:
For comparison, the tetrational tet to base \( e \) as obtained with Dmitriis Cauchy-integral algorithm is also drawn.
There is no hope to distinguish \( F_{2,3} \) and \( F_{4,3} \) 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 \( w(z)= F_{4,3}(z)-F_{2,3}(z) \), which is plotted with factor \( 10^{24} \), id est, \( y=10^{24}w(x) \), in dark pink at the bottom.
And even after so strong scaling-up, the difference \( w \) remains smaller than unity; it oscillates along the real axis, passing through zero at integer values of its argument and decaying at \( \pm \infty \).
Each of functions \( F=F_{L,d} \) satisfies the same equation \( F(x+1)=\sqrt{2}^{F(x)} \), at least in vicinity of the origen of coordinates and in the positive direction of the real axis. These functions are periodic, \( F_{L,d}(z+T_L)=F_{L,d}(z) \) for all \( z \) that belong to the range of holomorphism. The periods are \( T_2=2 \pi \mathrm{i}/\ln(\ln(2))\approx -17.143148179354847104 ~\mathrm{~i} \) and \( T_4=2 \pi \mathrm{i}/\ln(2 \ln(2))\approx 19.236149042042854712~\mathrm{~i} \), 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 \( F \) with modification of the argument: \( \mathcal{F}(x)=F(x+\theta(x)) \), where \( \theta \) is 1-periodic function, holomorphic at least in some vicinity of the real axis, such that \( \theta(0)=0 \). The mofigied function \( \mathcal{F} \) satisfies the same equation; \( \mathcal{F}(x+1)=\sqrt{2}^{\mathcal{F}(x)} \). However, while \( \theta \) is not identically zero, the modification destroys the periodicity of function, allthough the function \( \mathcal{F} \) may be also analytic and even entire (for the case \( L=4 \)). 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 \( F \) above are related: \( F_{2,3} \) can be expressed through \( F_{2,1} \) and \( F_{4,3} \) can be expressed through \( F_{4,5} \) with some complex constant ofsets of the arguments.
We consider the superexponentials on base \( b=\sqrt{2} \).
In addition to the upper superexponential (which is always bigger than the upper fixed point \( L=4 \) along the real axis) and the lower superexponential (which has values less than the lower fixed point \( L=2 \) along the real axis),
we plot and discuss also the two "intermediate" superexponentials with values between 2 and 4.
We denote the superexponentials as \( F_{L,d} \). The first subscript indicates the fixed point, used to develop the corresponding Schroeder function. For base \( \sqrt{2} \) 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, (\( L=2 \) and \( L=4 \)), we plot one function of each of these two classes, id est, four functions.
Additionally to the already introduced super-exponentials \( F_{2,1} \) and \( F_{4,5} \) we present here the super-exponentials \( F_{2,3} \) and \( F_{4,3} \). Which however are not distinguishable at the real axis:
For comparison, the tetrational tet to base \( e \) as obtained with Dmitriis Cauchy-integral algorithm is also drawn.
There is no hope to distinguish \( F_{2,3} \) and \( F_{4,3} \) 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 \( w(z)= F_{4,3}(z)-F_{2,3}(z) \), which is plotted with factor \( 10^{24} \), id est, \( y=10^{24}w(x) \), in dark pink at the bottom.
And even after so strong scaling-up, the difference \( w \) remains smaller than unity; it oscillates along the real axis, passing through zero at integer values of its argument and decaying at \( \pm \infty \).
Each of functions \( F=F_{L,d} \) satisfies the same equation \( F(x+1)=\sqrt{2}^{F(x)} \), at least in vicinity of the origen of coordinates and in the positive direction of the real axis. These functions are periodic, \( F_{L,d}(z+T_L)=F_{L,d}(z) \) for all \( z \) that belong to the range of holomorphism. The periods are \( T_2=2 \pi \mathrm{i}/\ln(\ln(2))\approx -17.143148179354847104 ~\mathrm{~i} \) and \( T_4=2 \pi \mathrm{i}/\ln(2 \ln(2))\approx 19.236149042042854712~\mathrm{~i} \), 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 \( F \) with modification of the argument: \( \mathcal{F}(x)=F(x+\theta(x)) \), where \( \theta \) is 1-periodic function, holomorphic at least in some vicinity of the real axis, such that \( \theta(0)=0 \). The mofigied function \( \mathcal{F} \) satisfies the same equation; \( \mathcal{F}(x+1)=\sqrt{2}^{\mathcal{F}(x)} \). However, while \( \theta \) is not identically zero, the modification destroys the periodicity of function, allthough the function \( \mathcal{F} \) may be also analytic and even entire (for the case \( L=4 \)). 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 \( F \) above are related: \( F_{2,3} \) can be expressed through \( F_{2,1} \) and \( F_{4,3} \) can be expressed through \( F_{4,5} \) with some complex constant ofsets of the arguments.