z^^z ?

[tommy1729]

I edited to change from "e" to "superE" on my post #5, sorry.

i dont know what your talking about actually.

There is this bifurcation base 1.6353... for the tetrational:

for b<1.6353... b[4]x has two fixpoints
for b=1.6353... b[4]x has one fixpoint
for b>1.6353... b[4]x has no fixpoint
on the positive real axis.

As you see, the bifurcation base 1.6353... of the tetrational corresponds to the bifurcation base \( e^{1/e} \) of the exponential.
(Also corresponds regarding other characterizations like the point b where b[4](b[4](b[4]...)) starts to diverge or the argument where the 4-selfroot is maximal)

The normal Euler constant e is now the one fixpoint of \( e^{1/e}[3]x \).
And the Super-Euler constant is the one (positive) fixpoint of \( 1.6353...[4]x \).

i suppose - without thinking about it - there is a fast converging method to find 1.6353... which also shows its transcendental.

i dont have much time to explore it nowadays ...

[Stan]

About the complex tetration thing...

I thought (intuitively) that:

e^^xi=cos(x)+sin(x)i

e^^2xi=cos(cos(x)+sin(x)i) + sin(cos(x)+sin(x)i))

e^^3xi=cos(cos(cos(x)+sin(x)i) + sin(cos(x)+sin(x)i)))
sin(cos(cos(x)+sin(x)i) + sin(cos(x)+sin(x)i)))

e^^kxi=cos(cos(cos(cos(x)+sin(x)i) + sin(cos(x)+sin(x)i)))
sin(cos(cos(x)+sin(x)i) + sin(cos(x)+sin(x)i)))) +....2^k times
sin(cos(cos(cos(x)+sin(x)i) + sin(cos(x)+sin(x)i)))
sin(cos(cos(x)+sin(x)i) + sin(cos(x)+sin(x)i))))...etc....

But this is MERELY an intuitive hunch.

Any thoughts on the matter?

[tommy1729]

that is completely wrong.

you should really learn more about the basics of elementary math functions and complex numbers if you want to follow or contribute to this forum.

also there is a difference between ^ and ^^ !