exp(x) - 1

[tommy1729]

this idea might have been posted before, forgive me if such is the case.


we know exp(x)-1 has a fixed point.

which leads to unique half-iterate and a somewhat unique superfunction.


so we might want to use the fixpoint of exp(x) - 1 for a ' surrogate fixpoint ' of exp(x).

here is how - if i dont blunder - :

using superfunction F(x) :

F(x + 1) = exp [ F(x) ] - 1.

which can be solved by taylor series i believe ?

now the simple but brilliant idea - if correct -

F( x + 1 ) + 1 = exp [ F(x) ]

generalize to

F ( x + a ) + a = exp exp exp ... a times [ F(x) ]


and Coo tetration follows !!?


regards

tommy1729