04/12/2021, 10:33 PM

As said before, I intend to use the incomplete Gamma function.

Let

We can compute this by infinite composition like ;

This is pretty easy and standard.

The idea to achieve approximate tetration would then be

, find .

then approximately.

However this is just an approximation and depends on the real part of which should preferably be a large positive real.

This leads to many ideas that may or may not work...

For instance assume the function is analytic in and let this real go to positive infinity.

OR find a larger ( larger real positive part preferably close to the real line ) solution and start from there.

or a sequence of .

Notice that tetration paths can self-intersect so this might just be other directions !

Another idea is the simple for integer m and start from there.

Ofcourse we choose m such that y_m has a large positive real part.

notice that m can also be negative and still be good !!

The problem here is when we pick different m for different x , we might end up in discontinu solutions.

A possible solution might be similar to the sequence above ; an infinite sequence of optimal m's.

The problem is mainly with the nonreals.

A few more comments

We could try the real line and then " simply " make analytic continuation.

This brings me to base change ideas or James Nixon's approach :

Let t go to +oo ;

to help find tetration.

Im not completely certain this is analytic but some comments might be useful.

For starters this is much more likely to be analytic then the base change or f(z) = ln .. ln ln ln 4^4^4^ ..4^z type methods.

1) How to take a single logaritm and have instant analytic continuation ?

PROPOSED SOLUTION

log(U(z)) = integral from a to c of U'(z) dz / U(z) + integral from c to b of U'(z) dz / U(z) + "constant".

Where a,b,c and "constant " are chosen wisely. That is to say : c = z , a is appropriate and c is to avoid division by zero.

the path a,b,c is analytic.

( assuming we already have analytic continuation of U(z) )

example

log(exp(z)) = integral from 1 to z/2 of exp'/exp + integral from z/2 to z of exp'/exp + constant = z.

**

2) for real s > 1 how to bound ln ln ln ... exp(a1* exp( a2 * exp ( ... ?

If you want to bound compositions of positive real direction from above you simply take weakest ones first.

for instance for a,x > 1 ; exp(a x) is between a * exp(x) and exp(x)^a.

so :

ln ln ln ... exp(a1* exp( a2 * exp ( ... < s^(a1*a2*a3*...)

This automatically proves that converges for real y > 1 and is bounded by y^(a1*a2*a3*...).

***

Combining 1) and 2) might help in proving that the proposed solution is analytic ??

***

There is more to say but James has already done so.

***

Many more ideas are in my head but they are complicated and doubtful.

I just wanted to share some easy ideas here.

I might echo some ideas of James.

***

The idea of chaos leads to the fear that slightly different bases than e lead to chaos and hence log(0) for the Jn(z) for nonreal z.

The idea occurs that when the bases are close enough to e , than all is fine.

Which leads me to ideas like : is this gamma fast enough ??

Or should we have tetrational growth ??

Another idea is this : Can speed be too fast ???

I mean if it is too fast our function might be to close to a finite power tower because the tail goes to zero too fast ??

This might loose analyticity or some undefined " smoothness or uniqueness criterion " .

***

Also Im not aware of an efficient way to avoid overflow to compute things like ln...ln ln ln 4^4^4^..^4^z.

Precompute taylor series or carleman matrices seems the only way but that is not so efficient.

funny because it converges fast !

***

What do you think ?

regards

tommy1729

Tom Marcel Raes

Let

We can compute this by infinite composition like ;

This is pretty easy and standard.

The idea to achieve approximate tetration would then be

, find .

then approximately.

However this is just an approximation and depends on the real part of which should preferably be a large positive real.

This leads to many ideas that may or may not work...

For instance assume the function is analytic in and let this real go to positive infinity.

OR find a larger ( larger real positive part preferably close to the real line ) solution and start from there.

or a sequence of .

Notice that tetration paths can self-intersect so this might just be other directions !

Another idea is the simple for integer m and start from there.

Ofcourse we choose m such that y_m has a large positive real part.

notice that m can also be negative and still be good !!

The problem here is when we pick different m for different x , we might end up in discontinu solutions.

A possible solution might be similar to the sequence above ; an infinite sequence of optimal m's.

The problem is mainly with the nonreals.

A few more comments

We could try the real line and then " simply " make analytic continuation.

This brings me to base change ideas or James Nixon's approach :

Let t go to +oo ;

to help find tetration.

Im not completely certain this is analytic but some comments might be useful.

For starters this is much more likely to be analytic then the base change or f(z) = ln .. ln ln ln 4^4^4^ ..4^z type methods.

1) How to take a single logaritm and have instant analytic continuation ?

PROPOSED SOLUTION

log(U(z)) = integral from a to c of U'(z) dz / U(z) + integral from c to b of U'(z) dz / U(z) + "constant".

Where a,b,c and "constant " are chosen wisely. That is to say : c = z , a is appropriate and c is to avoid division by zero.

the path a,b,c is analytic.

( assuming we already have analytic continuation of U(z) )

example

log(exp(z)) = integral from 1 to z/2 of exp'/exp + integral from z/2 to z of exp'/exp + constant = z.

**

2) for real s > 1 how to bound ln ln ln ... exp(a1* exp( a2 * exp ( ... ?

If you want to bound compositions of positive real direction from above you simply take weakest ones first.

for instance for a,x > 1 ; exp(a x) is between a * exp(x) and exp(x)^a.

so :

ln ln ln ... exp(a1* exp( a2 * exp ( ... < s^(a1*a2*a3*...)

This automatically proves that converges for real y > 1 and is bounded by y^(a1*a2*a3*...).

***

Combining 1) and 2) might help in proving that the proposed solution is analytic ??

***

There is more to say but James has already done so.

***

Many more ideas are in my head but they are complicated and doubtful.

I just wanted to share some easy ideas here.

I might echo some ideas of James.

***

The idea of chaos leads to the fear that slightly different bases than e lead to chaos and hence log(0) for the Jn(z) for nonreal z.

The idea occurs that when the bases are close enough to e , than all is fine.

Which leads me to ideas like : is this gamma fast enough ??

Or should we have tetrational growth ??

Another idea is this : Can speed be too fast ???

I mean if it is too fast our function might be to close to a finite power tower because the tail goes to zero too fast ??

This might loose analyticity or some undefined " smoothness or uniqueness criterion " .

***

Also Im not aware of an efficient way to avoid overflow to compute things like ln...ln ln ln 4^4^4^..^4^z.

Precompute taylor series or carleman matrices seems the only way but that is not so efficient.

funny because it converges fast !

***

What do you think ?

regards

tommy1729

Tom Marcel Raes