f( f(x) ) = exp(x) solved ! ! !

[tommy1729]

On sci.math i posted the solution to f( f(x) ) = exp(x).

That is of course important towards tetration.

Here is the post :

http://mathforum.org/kb/thread.jspa?threadID=1891059

Its important to note that the same strategy works to compute e.g.

f( f( f(x) ) ) = exp(x)

And can thus be used to compute tetration on the real line numerically.

Here is the post explicitly , in case sci.math is not accessible ( which happens sometimes ) :

brute force tetration / greedy tetration
Posted: Jan 27, 2009 9:39 AM Plain Text Reply


here i introduce a brute force / greedy algoritm to compute f(f(x)) = exp(x).

its works best outside the unit radius.

it is based upon the approximation below , and is trivial considering the approximation.

since , once we have a sequence of ever better getting approximations , taking that limit gives the desired result.

the approximations are also usefull because of computational boundaries.

the algoritm is the limit n -> oo

note that f(f(x)) = g(x) can be computed if g(0) = 0


f(f(x)) = exp(x) - exp( -1 * 25^n * x^2 )

examples :

n = 1 :

f(f(x)) = exp(x) - exp( -25 * x^2 )

n = 2 :

f(f(x)) = exp(x) - exp( - 625 * x^2 )

n = 3 :

f(f(x)) = exp(x) - exp( - 15625 * x^2 )

...

i , tommy1729 , am the first to invent this.

i will not accept shameless copies of this without mentioning me.

copyright tommy1729


regards

tommy1729

( end quote )

I am the sole inventor of this.

regards

tommy1729

" Statisticly , i dont exist " tommy1729