complex iteration (complex "height")

[bo198214]

That you believe me, here is the regular \( \frac{1}{e}[4]t \) for \( t \) in the range of -1.5..5, the value 0 is of course taken for \( t=-1 \).
   

[Ivars]

I see. And a purely imaginary value as well somewhere between -1 and -1.5.

Now it gets too fast for me, may be You can explain how it all comes together when You feel like it. It is really exciting.


I was wondering if such spirals can form some basis for expansion of complex functions, maybe resulting from tetration, in series , similar to Fourier (not that I know much about it, but spiral basis would seem so natural for many things happening in nature).

Ivars

[Ivars]

Since I still can not iterate complex numbers, let me make a few opportunistic wild quesses which look nice

\( {1/e}[4]{-\Omega}=\Omega+I*(1-\Omega)=0.567143+I*0.432857... \)

\( {1/e}[4]{-1/\Omega} = \Omega+ I*{(1/\Omega}-1)=0.567143+I*0.763224.. \)

\( {1/e}[4]{\Omega}=\Omega+I*\Omega^{1/(1-\Omega)}=0.567143+I*0.26975. \)


If one of them is close, even real or imaginary part?.

Ivars

[bo198214]

Since I still can not iterate complex numbers, let me make a few opportunistic wild quesses which look nice

\( {1/e}[4]{-\Omega}=\Omega+I*(1-\Omega)=0.567143+I*0.432857... \)

\( {1/e}[4]{-1/\Omega} = \Omega+ I*{(1/\Omega}-1)=0.567143+I*0.763224.. \)

\( {1/e}[4]{\Omega}=\Omega+I*\Omega^{1/(1-\Omega)}=0.567143+I*0.26975. \)


If one of them is close, even real or imaginary part?.

No, quite wrong guesses:
\( \begin{align*}
\frac{1}{e}[4](-\Omega)&=0.4035614-0.4516711*I\\
\frac{1}{e}[4]\frac{-1}{\Omega}&=1.0316317+1.0778810*I\\
\frac{1}{e}[4]{\Omega}&=0.4921705+0.2519841*I
\end{align*} \)

[Ivars]

Henryk,

May be You could send me/publish here a table of values? I was trying to provoke you...

I am almost sure the non-integer part of real positive/negative iterations will hide some numerical surprises which will allow to establish (perhaps) some other analytical links in some parts of the spiral then pure numerical iteration according to Your GREAT formula.

And if not,it will show we should look into complex heights, or elsewhere ( pentation etc).

I still can not help being more impressed by your discovery than I am able to utilize it, but its only few days old, so there is still time. There must be heaps of insights.


BTW, if You go back more with negative heights, like -2, You enter positive real values of result , while -3 leads to negative etc, 4 to positive, -5 to negative etc. I think I mentioned somewhere that such oscillations may happen at least for h(i^(1/i) and h(-1^(1/-1)) as +- i and +- 1 as powers of i and -1 are used (i^n^(1/i^n)), (-1^n) ^(1/(-1)^n)) . Perhaps this spiral must be placed orthogonally to\( b=a^{(1/a)}[4]infinity \) value =a in every point of the plane in Gottfrieds spider graph?

Or attached to every such point in another 2D space....giving the 4D space of h(z^(1/z))=z. Excuse me , I am rambling a little, at work, no time to think carefully.


Ivars