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I got inspired by this

http://math.eretrandre.org/tetrationforum/showthread.php?tid=1149

Where Tommy assumes problems for tetration.

I was intrested in a closed form for when the fixpoint is on the edge of a new branch of ln or equivalent on the edge of the univalent zone of the exponential.

So i write ( with $a,q$ real variables and $q>1$)

$$ exp( q ( a + \frac {2 \pi}{q} i) ) = a + \frac{2\pi }{q} i$$

This reduces

$$ exp(q a) = a + \frac {2 \pi }{q} i ?? $$

Or maybe

$ exp ( q a) = a $ ?

Im confused.

Maybe im confused because of branches.

What about closed forms for $a,b,q$ ?

Does that fixpoint even exist ??

How does the LambertW function handle this ?

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    $\frac{2\pi}{qi}$ or $\frac{2\pi}{q}i$? – 2017-02-04
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    I edited. @Aweygan – 2017-02-04
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    I don't see the problem in the first question. If for some $a,q$ we have with $x=(a+{2 \pi i \over q })$ the identity $x=\exp(q(a+{2 \pi i \over q }))$ then we have also $x = \exp(q \cdot a)=$ $\exp(q(a+{2 \pi i \over q }))=$ $\exp(q(a+{4 \pi i \over q }))= ... $ – 2017-02-05

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