# Tetration Forum

Full Version: simple equation for R -> R
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there is a simple equation to find a real to real tetration.

( [0,oo] to [0,oo] , no negative reals of course )

it works for all solutions.

lets call the solution 'kneser'.

and set kneser(0) = 0 , sexp(0) = 0.

real(z) = x

im(kneser(z)) = 0

sexp(x) = kneser(z)

thats it !

however finding the solution is something else.

i assume that by riemann mapping theorem that 'this sexp' is coo if ' this kneser ' is.

finding the parametric real and im part of ' kneser ' might be as complicated or even equivalent to riemann mapping.

and i assume the coo prop of the parametric real and im part is equivalent to the coo prop of ' kneser ' or ' sexp '.

looking at it this way , its trivial to see that riemann mapping , fourier series and the 'wave' are strongly related , not to say equivalent.

the question is if the equations

set kneser(0) = 0 , sexp(0) = 0.

real(z) = x

im(kneser(z)) = 0

sexp(x) = kneser(z)

lead to new insights or alternative methods.

i propose that the curve of the 'knesers' ( the parametric wave of the real outputs of all known non R->R solutions ) get a special function name or special command name ( for general iterations ).

( command could be 'knesercurve(f(x))' and in the general case its very likely that the curve cannot be computed if the superfunction cant , but who knows ! )

i wanted to note that the 'wave' does have some limitations and does not have total freedom as sometimes believed :

it must be an analytic wave and it must be a bounded wave that converges on C ( luckily for four series ! and thus not double periodic or finite radius )

but thats not all.

since we have a smooth curve the equations presented might be solvable without riemann mapping !

since we have a smooth curve we might use newton iteration.

im convinced that an iterative system exists to solve such a problem that is hardly more complicated than computing a schroeder function.

that is however a numerical method not suited for closed form ( unless the algoritm itself ( limit ) but prob no sum or integral ) or properties or proofs , unless by the squeezing theorem and comparing different tetration solutions...

i guess we are pretty close to reunderstanding knesers solution and being able to compute it ( rather than just the non-constructive existance ).

i know im not the only one doing research on it ( e.g. sheldonison ) , but i didnt want this to be left behind.

and maybe i helped some people who didnt understand kneser or sheldon and do understand now.

but im not finished...

because of kneser(z+1) = exp(kneser(z))

it is clear that the kneser curve has period 1.

kneser(0) = 0 and kneser(1) = 1.

if we draw a straith line trough them , then i assume the curve must be above or below the line and never cross it.

is that true ?

regards

tommy1729
ok , i must say i did not mention my assumption that the ' theta wave ' ( kneser curve ) is "smooth looking".

the shape of the theta wave is important ...

i assumed for all tetration solutions not yet mapping R -> R , that the theta wave is coo apart from the numbers [0,1,b,b^b,...] and looking " gauss-like ".

with gauss-like i mean that it looks like the error function , one single max value in the middle , symmetric , no local minimum.

for such a shape simple methods may be usefull.

such as newton iterations. at least for numerical solutions. on the other hand we are not just looking for 'a' tetration , but 'the' tetration.

regards

tommy1729
(08/11/2010, 04:10 PM)tommy1729 Wrote: [ -> ].... finding the parametric real and im part of ' kneser ' might be as complicated or even equivalent to riemann mapping.

and i assume the coo prop of the parametric real and im part is equivalent to the coo prop of ' kneser ' or ' sexp '.
I freely admit that my higher mathematics education has been stretched way beyond its limits. My wife thinks I should go back to school, so I can get an advanced degree and teach at a University. Anyway, all that aside. I don't have a really good background in understanding the Riemann mapping theorem, this despite having stumbled upon an algorithm to generate the Riemann mapping for tetration. I gather from reading on the internet, that the constructive Riemann mapping approaches are matrix based, and very slowly converging in the case of a singularity (which of course, is our case).

I would assume that except at the singularity, one could continue the function, and that the theta(z) at the real axis would have to be analytic, coo.

Quote:....
but im not finished...

because of kneser(z+1) = exp(kneser(z))

it is clear that the kneser curve has period 1.

kneser(0) = 0 and kneser(1) = 1.

if we draw a straith line trough them , then i assume the curve must be above or below the line and never cross it.

is that true ?

regards

tommy1729
I'm not sure if by Kneser curve you mean sexp(z), which does have one inflection point somewhere near -0.5.

If you're talking about the theta(z) function, it has a really really bad singularity. As z superexponentially approaches the singularity at integer values, the behavior gets worst and worst, than the plots I posted along with the equations. The imag(z) oscillates, going up and down, and the real(z), which looks like it might converge to a value, instead super-logarithmically oscillates, growing to +real infinity, while oscillating in the imag(z)! I'll plot an updated post some time, along with some comments.
- Sheldon