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 Searching for an asymptotic to exp[0.5] tommy1729 Ultimate Fellow Posts: 1,372 Threads: 336 Joined: Feb 2009 09/06/2016, 03:47 PM Back to basics In addition to post 17,18 notice that D exp^[1/2](x) = D exp^[1/2] (exp(x) ) * exp(x) / exp^[1/2](exp(x)). That follows from ln exp^[a] ( exp(x) ) = exp^[a](x) and the chain rule for derivatives. By induction / recursion this gives a nice way ( product ) to compute the derivative. This strenghtens the conclusions from post 17 , 18 and shows that 1 + o(1) <<_n 2. ( smaller after only a few iterations n ) We conclude by noting that the Taylor T T = Sum_{K=4}^{oo} d_k x^k With d_k = exp( - k^2 ) grows slower then exp^[1/2](x) , yet faster then any polynomial. T has growth 0 , like exp( ln^2 (x) ) and similar. Did we meet T yet ?? I believe T was a fake of Some elementairy like exp( ln^2 ) or such ... This again leads to the desire of inverse fake or its related integral transforms ... --- Regards Tommy1729 tommy1729 Ultimate Fellow Posts: 1,372 Threads: 336 Joined: Feb 2009 03/15/2018, 01:23 PM I request complex plots of f(x) = fake exp^[1/3](x) , f(f(x)) and f(f(f(x))). Like sheldon did for fake exp^[1/2](x) in one of the early posts in this thread. It is very important !! ( potentially new results/conjectures based on those plots ! ) Regards Tommy1729 Ps make sure to make backups of this website/content ? Bo ? « Next Oldest | Next Newest »

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