### Half-iterate of exp(x).

 We consider 2 * sinh(x).   We take the half-iterate of it by using taylor series expanded at 0.Thus f(x) is a taylor series , f(f(x)) = 2*sinh(x).Clearly f(x) is then a good asymptote for the half iterate of exp(x) for large x.Let g(g(x)) = exp(x)Then g(x) =Lim k-> oo   log log log log ...( k times ) (f( e^e^e^...( k times ) ^ x))   Clearly this can be generalized to tetration.   For a real 'r' to compute exp^[r] ( ^[] is iteration ) replace f(x) with (2*sinh(x))^[r] in the above formula.   Now you have ( for x and r being real ) exp^[r](x) aka tetration. It turns out this function is nowhere analytic (for the variable x) but it is continuous and infinitely differentiable on the real axis (C^oo for fixed r and variable x). By analogue for a realvalued b we can compute (b^(x))^[r] and find the same properties as long as b > exp(1/2).  All of this has been formally proven for b > exp(2) but Im working on the smaller bases too. One of the main reasons is that for b > exp(2) , 2*sinh(2x) has only one fixpoint (=0) whereas for exp(1/2) < b < exp(2) you have 3 fixpoints : zero and two nonreal complex conjugates. Below is a link to my profile at the tetration forum. At the moment I only accept the tetration forum as a good source for tetration although you can find some info and links on wiki , I do however not accept the wiki page. http://math.eretrandre.org/tetrationforum/member.php?action=profile&uid=47   Thank you for your intrest. tommy1729

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