Multiple exp^[1/2](z) by same sexp ?

[tommy1729]

Consider the analytic solution sexp(z) that has the minimal amount of singularities => only singularities on the negative real axis.

( uniqueness has been shown , related is TPID 4 : http://math.eretrandre.org/tetrationforu...hp?tid=747)

Now let sexp(A) = u where Re(A) > 0,Im(A) > 0 and exp^[v](u) = u for real positive v.

Notice that most u have such a v because of the chaotic behaviour of the iterations of exp. ( we ignore branches and functional equations in this sentense )

Now if such A,u are defined on the fundamental branch of sexp such that we have sexp(z+1)=exp(sexp(z)) then we have to conclude something special :

exp^[v](u)= u = sexp(A) = sexp(A+v)

Now if all if for w and v : 0 < w < 1 << v

exp^w(u) = sexp(A + w) = sexp(A + v + w) = sexp(A+ 2v + w)

THEN WE MUST CONCLUDE BY INDUCTION AND ANALYTIC CONTINUATION :

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sexp is periodic with period v !!
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But this contradicts the nonperiodicity of sexp and the branches of sexp !!

Hence we must have for some u : set w = 1/2

then sexp(A+1/2 + v) =/= sexp(A+1/2)

THUS

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exp^[1/2](u) IS NOT DEFINED UNIQUELY BY THE SAME BRANCH OF SEXP THAT INCLUDES A
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EUREKA !

this is surprising not ?

this reminds me of the recent threads such as "slog(sexp(z)" AND MUCH MORE of a thread by gottfried :


http://math.eretrandre.org/tetrationforu...hp?tid=499

I bet you can see the relevance.

regards

tommy1729