11/06/2008, 03:19 PM
(This post was last modified: 11/10/2008, 09:47 AM by Kouznetsov.)
Lemma is not yet ready, but Bo asked me about pics of slightly modified tetration.
I consider the simplest possible modification of tetration
\( \mathrm{tem}(z)=\mathrm{tet}(J(z)) \)
where \( J(z)=z+a\cdot\sin(2 \pi z) \)
Such modified tetration satisfies the tetration equaitons,
\( \mathrm{tem}(z+1)=\exp(\mathrm{tem}(z)) \),
\( \mathrm{tem}(0)=1 \)
For \( a=10^{-9} \), the function \( J \) is plotted in the complex plane
It is almost identical function, but the defiation is seen, if the imaginary part becomes of larget than 3.
Now, the plot of modified tetration:
The grid shown occupies the range [-10,10], [-4,4] with step unity.
Levels of integer values of \( \Re(\mathrm{tem} \) and
Levels of integer values of\( \Im(\mathrm{tem} \) are plotted.
Below I show the zoom-in of the part of the previoous figure:
There are cutlines there. The structure in the upper part folloes the topology of tetration, but it is reduced in size and strongly deformed.
For highest harmonics like \( \sin(4\pi z) \) or \( 1\!-\!\cos(4\pi z) \), added to the argument of tetration in definition of mofified tetratio "tem", the
structures of cuts is smaller, denser and approach closer to the real axis.
If some function \( F \) satisfies equations
\( \mathrm{tem}(0)=1 \)
\( \mathrm{tem}(z\!+\!1)=\exp(\mathrm{tem}(z)) \forall z : |\Re(z)|<1, |\Im(z)|<4 \),
and for some \( x\in \mathbb{R} : -1\!<\!x\!<\! 0 \), the function differ from tetration for at least \( 10^{−9} \), id est,
\( |Fxz)\!-\!\mathrm{tet}(x)| > 10^{-9} \).
then function \( F \) is not holomorphic in
\( \{z\in \mathbb{C}: |\Re(z)|<1, |\Im(z)|<4 \} \)
In such a way, any deformation of tetration (even small) breaks its continuity.
See also
http://en.citizendium.org/wiki/Tetration
I consider the simplest possible modification of tetration
\( \mathrm{tem}(z)=\mathrm{tet}(J(z)) \)
where \( J(z)=z+a\cdot\sin(2 \pi z) \)
Such modified tetration satisfies the tetration equaitons,
\( \mathrm{tem}(z+1)=\exp(\mathrm{tem}(z)) \),
\( \mathrm{tem}(0)=1 \)
For \( a=10^{-9} \), the function \( J \) is plotted in the complex plane
It is almost identical function, but the defiation is seen, if the imaginary part becomes of larget than 3.
Now, the plot of modified tetration:
The grid shown occupies the range [-10,10], [-4,4] with step unity.
Levels of integer values of \( \Re(\mathrm{tem} \) and
Levels of integer values of\( \Im(\mathrm{tem} \) are plotted.
Below I show the zoom-in of the part of the previoous figure:
There are cutlines there. The structure in the upper part folloes the topology of tetration, but it is reduced in size and strongly deformed.
For highest harmonics like \( \sin(4\pi z) \) or \( 1\!-\!\cos(4\pi z) \), added to the argument of tetration in definition of mofified tetratio "tem", the
structures of cuts is smaller, denser and approach closer to the real axis.
If some function \( F \) satisfies equations
\( \mathrm{tem}(0)=1 \)
\( \mathrm{tem}(z\!+\!1)=\exp(\mathrm{tem}(z)) \forall z : |\Re(z)|<1, |\Im(z)|<4 \),
and for some \( x\in \mathbb{R} : -1\!<\!x\!<\! 0 \), the function differ from tetration for at least \( 10^{−9} \), id est,
\( |Fxz)\!-\!\mathrm{tet}(x)| > 10^{-9} \).
then function \( F \) is not holomorphic in
\( \{z\in \mathbb{C}: |\Re(z)|<1, |\Im(z)|<4 \} \)
In such a way, any deformation of tetration (even small) breaks its continuity.
See also
http://en.citizendium.org/wiki/Tetration