complex iteration (complex "height")

[bo198214]

Not afraid, but difficult for me to work with before I learn enough software how to handle complex number iterations

Ya right, probably excel can not compute with complex numbers ...

So You mean Your formula works at least in the range \( e^{-e}; e^{1/e} \) also below 1? That is wonderful achievement

Yes, exactly, isnt it?! Bases below 1 were what *I* was afraid of!

For base \( {1/e} \), \( {1/e}[4]-{1/\Omega} =1 \) - you can add this to your graph as well perhaps another spiral will emerge which should pass via this point?

You mean \( \frac{1}{e}[4](-\frac{1}{\Omega})=1 \)? Thats not true. \( b[4]t \) is only real for integer \( t \) (considering \( e^{-e}<b<1 \)).

Your nice spiral graph (see-spirals are involved) anyway converges to my beloved \( \Omega \), as it should as \( {1/e}[4]\infty=\Omega=0.567143.. \).

Exactly: \( a^{1/a}[4]\infty=a \).

But what about base \( e^{1/e} \) which is limit case?

There are also iteration formulas, I think you can find them in Ecalle's work. However if I remember right, its not one formula, but some intermediate steps required. Meanwhile however you can just consider it as a limit of the other case.