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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 |