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+--- Thread: Inverse power tower functions (/showthread.php?tid=1046)
Inverse power tower functions - tommy1729 - 01/04/2016
I consider the functional inverse of power towers made with x and 2.
These functions are imho fundamental ; they are the seed for brute asymptotes to most real-entire functions.
Height 2
Inv x^2
Inv 2^x
Inv x^x
With their resp solutions : sqrt , binairy log , ssqrt.
Notice the ssqrt can be expressed by ln and lambertW.
Most strictly rising functions that grow slower then x already grow like logs , powers and ssqrt type functions ( by finite composition , addition and product ).
I came to consider the enumeration of these functions.
For height 2 , as shown above , we have 3 functions.
The pattern seems simple , but might not be.
How many functions do we have Up till height 3 ?
You probably guessed 6 if you are fast.
But it is 5.
Because
Y = x^ ( x^2)
Ln(y) = ln x x^2.
This equation can be solved by LambertW ( just like x^x = y could ).
So we get 3 for height 2 and max 2 extra for height 3 :
Inv x^x^x
Inv x^2^x
Giving a Total of max 5.
The fact that x^x^2 reduces is domewhat surprising.
So care is needed.
So for instance how many fundamental function do we have Up to height 17 ?
A related question is how Some of them make good asymptotics of others and Some do not.
For instance ssqrt has asymptotics in terms of the others of height 2 ( logs and powers ).
Conjectures are easy to make for the amount of functions Up to Some height
For instance
5 + 2^(h-3) for hights h >=3.
But hard to prove.
Regards
Tommy1729