Has anyone studied the iterates of \(\frac{1}{e^{-1/z}}\)

[Daniel]

Has anyone studied the iterates of \( \frac{1}{e^{-1/z}} \)?

\[\frac{1}{e^{-1/z}}=e^{1/z}\]
\[\frac{1}{e^{-\frac{1}{e^{-1/z}}}}=e^{e^{1/z}}\]
\[\frac{1}{e^{-\frac{1}{e^{-\frac{1}{e^{-1/z}}}}}}=e^{e^{e^{1/z}}}\]

[Daniel]

I am working on using the exponential function's fixed point at \(\infty\) in an attempt to define tetration \(^z a\) for \(a>e^{1/e}\). Generalizing \(e^{-\frac{1}{z}}\) to \(g(z)=1/f{(\frac{1}{z})}\) where \(f(\infty)=\infty\).

The iterates of \(g(z)\) are,
\[\frac{1}{f\left(\frac{1}{z}\right)},\frac{1}{f\left(f\left(\frac{1}{z}\right)\right)},\frac{1}{f\left(f\left(f\left(\frac{1}{z}\right)\right)\right)},\frac{1}{f\left(f\left(f\left(f\left(\frac{1}{z}\right)\right)\right)\right)}\]

[tommy1729]

Has anyone studied the iterates of \( \frac{1}{e^{-1/z}} \)?

\[\frac{1}{e^{-1/z}}=e^{1/z}\]
\[\frac{1}{e^{-\frac{1}{e^{-1/z}}}}=e^{e^{1/z}}\]
\[\frac{1}{e^{-\frac{1}{e^{-\frac{1}{e^{-1/z}}}}}}=e^{e^{e^{1/z}}}\]

Im sorry but iterating exp(1/z) I get

\[e^{1/z}\]
\[e^{e^{-1/z}}\]
\[e^{e^{-e^{-1/z}}}\]

etc

So I would like to point out :

1) your result is wrong.
2) the fixpoint at oo of exp(x) is not really an analytic fixpoint. In fact exp(oo) takes on any value, even 0.
exp(x) - x = 0 does not even have a fixpoint at positive infinity in the sense that lim exp(x) - x > 0.
3) This looks alot like using base -e ( MINUS e ! ).
This base is thus below the sheltron region rather than above.

iterating exp(1/z) =?=

exp^[s]( exp_{-e}^[f(s)]( (-1)^g(s) 1/z ) )

for some f(s) and g(s) ??? hmm

4) Maybe we should go backwards :

iterate ln(z)^(-1)

( the inverse of exp(1/z) )

or the general case ln(z)^(t)

Notice ln(z)^(-1) has the fixpoint

ln(z)^(-1) = z

=>

ln(z) = 1/z

=>

z ln(z) = 1

z= exp( LambertW(1) )

( about 1.7632.. )

---

Now if we differentiate the fixpoint of 1/ln(z) we get

1/( z ln(z)^2 )

plugging in the fixpoint exp(W(1) ) we get

1/ ( exp(W(1)) W(1)^2 )

= 

1/( 1 * W(1) )

=

W(1)^(-1)

=

also exp(W(1)) !! 

( about 1.7632.. )

so the fixpoint has its own derivative cool.

***

other bases or log(x)^2 could also be considered.

But maybe you do not want to go there ?


Regards

tommy1729