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x exp(x) musing
#1
let f( x exp(x) ) * exp(x) * (x+1) = f(x)

then F( x exp(x) ) = F(x) + C

inversing ->

G(x+C) = G(x) * exp(G(x))

now notice G(x + n C) =
G(x) * exp(G(x)) * exp(G(x))^exp(G(x)) * ... n times.

thus exp(G(x)) ^^ n = G(x + n C) / G(x + n C - C) ( Form 1 )

and we have a solution for tetration.

now that was very briefly the idea , i know alot relates to it.

another related idea is to take the integral ;

because sexp'(x) = sexp(x) * sexp(x-1) * sexp(x-2) * ... * C2.

so sexp(x) relates to integral G(x). ( " Form 2 " not a formula yet actually :p )

now we have 2 strongly related ideas.

and keep in mind that our base needs to be bigger than eta.

for real bases > eta ; Form 1 or Form 2 might apply to one of the solutions for tetration ... ( or both sometimes !? since Derivate = prod )


i know , i know , ive only given ideas and equations , not solutions or proofs.

but still , i think it is worth consideration.

and maybe f(x) can be found by current attempts for tetration.

and G(x) might be found by inversing the ( modified ) carleman matrix of F(x)


notice also that x exp(x) has a unique real fixed point !!

i used 'x' instead of 'z' because mainly we want R > eta -> R

btw G(x) should be strictly increasing on the reals.


i hope that idea was not retarded Smile

regards

tommy1729
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Messages In This Thread
x exp(x) musing - by tommy1729 - 07/01/2009, 03:19 PM
RE: x exp(x) musing - by BenStandeven - 07/03/2009, 03:40 PM
RE: x exp(x) musing - by tommy1729 - 07/03/2009, 07:34 PM
RE: x exp(x) musing - by BenStandeven - 07/04/2009, 11:25 PM
RE: x exp(x) musing - by bo198214 - 07/05/2009, 07:12 PM
RE: x exp(x) musing - by tommy1729 - 07/05/2009, 07:33 PM

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