Dmitrii Kouznetsov's Tetration Extension

[bo198214]

I have plotted the only one, \( F \) such that \( F(0)=1 \).
At \( b=\sqrt{2} \), for example,
\( \lim_{x \rightarrow \infty} F_{\sqrt{2}}(x+{\rm i}y)=2 \);
\( \lim_{x \rightarrow -\infty} F_{\sqrt{2}}(x+{\rm i}y)=4 \).

This contradicts \( F(z^\ast)=F(z)^\ast \)?

No. \( F_{\sqrt{2}}(x+{\rm i}y) \) approaches the limiting values at any real \( y \). (in spite the cutlines!)

Oh that was my mistake, I thought you are speaking about the limits for imaginary infinity.

I tried to post the code as "source" together with the picture, but it was not accepted...

Ya, give me the extension of the file so I can add it to the accepted file types. Otherwise you just can append a .txt to the end of the filename, then its accepted if not too big. Suffix .gz also always works up to 1MB.

I know that usually the "self-made" codes dislike to run at another computer, so, let us do it with few steps, reproducing the figures one by one. Let us begin with very simple code. Please, reproduce first Figure 1 from the source posted at http://en.citizendium.org/wiki/Image:Exa...nLog01.png
please, tell me how does it run at your computer and send me (or post) the resulting picture. It is supposed to be <b>identical</b> with the eps file I got at my computer.

Gosh, you are really doing that in C! Creating eps in C, its unbelievable. But yes the code compiles with -lm and produce the same (at least visually) image.

Yes, but they seem to be singular.. What example would you suggest to begin with?

I would begin with what I am best with: tetration by regular iteration.

"Equal to mine" is tetration that has no singularities at the right hand side of the complex halfplane.

But only if your uniqueness conjecture is correct.
Ok this demand can reduced to having no singularities in the strip \( S \) of \( 0<\Re(z)\le 1 \).

I did not see any map of real and imaginary parts, nor those of modulus and phase. Tell me if I am wrong.

Hm, indeed I think we have only real plots available, as I am not familiar with complex function visualization. However most tetrations considered here are given as a powerseries developed at 0. So its not so easy to continue them to the whole strip \( S \). However for the regular tetration I have an iterative formula so this should be possible to make a plot similar to yours. And as I already explained, it is periodic along the imaginary axis, hence has no limit at \( i\infty \). Does that mean that it is different to your's? Which value does your \( F_{\sqrt{2}} \) assume at \( i\infty \)? Is it the nearest complex fixed point of \( \sqrt{2}^z \)?