(08/05/2021, 04:51 PM)Leo.W Wrote: Hey James,

I've lately been considering an anologous familt of asymptotic tetration functions, satisfying the recurrence

with arbitrary constant a and b

I simply take and solve term by term in series, which uses this code I wrote by wolfram mathematica 12, it iterates the recurrence to get the function converged

I think maybe there's a relation between these functions, especially seeing how it diverges when a is close to 1, so I think, if this is correct:Code:`Clear[A, B, term, aa, IS, Z]`

(* Solving \[Alpha] in coefficients *)

A = 1 + I;

B = 1/2;

term = 15;

aa[0] = 0;

aa[1] = 1/A;

IS = 1/A xx + Sum[aa[n] xx^n, {n, 2, term}];

Z = xx Series[Exp[IS Log[B]], {xx, 0, term}] - (IS /. xx -> A xx);

For[i = 2, i <= term, i++,

temp = Solve[Coefficient[Z, xx, i] == 0, aa[i]];

aa[i] = Simplify[temp[[1, 1, 2]]]]

Clear[\[Alpha], \[Beta], ff]

\[Alpha][z_] := Sum[aa[n] z^n, {n, 0, 15}]

\[Beta][z_] := Module[{x, q, o},

x = N[z, 200];

x = SetPrecision[x, 200];

q = 0;

While[Abs[x] > 10^-50,

x = x/A;

q = q + 1];

o = \[Alpha][x];

While[q > 0, o = x B^o; q = q - 1; x = A x];

Return[o]] /; Abs[A] > 1

\[Beta][z_] := Module[{x, q, o},

x = N[z, 200];

x = SetPrecision[x, 200];

q = 0;

While[Abs[x] > 10^-50,

x = x A;

q = q + 1];

o = \[Alpha][x];

While[q > 0, o = Log[B, o/x]; q = q - 1; x = x/A];

Return[o]] /; Abs[A] < 1

ff[z_] := \[Beta][A^z] /; Abs[A] > 1

ff[z_] := \[Beta][A^(z - 1)] /; Abs[A] < 1

and g(a) is only determined by a, exploding to infinity when a is getting closer to 1

Also, these functions are multivalued(Taken the relation between f(z) and f(z-1)), maybe associated with Riemann surface?

Leo

Hey, Leo

I'm sorry; I don't think I follow. Would you mind elaborating? What is , particularly?

Regards, James

OHHHH WWAIT, nevermind, I get it. You meant to write . You are absolutely correct.

What you have constructed here; using Sheldon's idea of a modified Schroder function; you've made,

This function will be holomorphic for . This is similar to how I constructed the method, where I took . The conjecture that stands is that this can only construct a tetration on . And converges nowhere in when you apply iterated logs.

You're construction method is perfectly valid though; it's how Sheldon justified my method; both ways are equivalent; his is more hands on with taylor series though.

By this, I mean, you can construct a family of tetrations,

It will probably diverge in though. It's going to look like the method.

I'd suggest looking at something that solves the asymptotic equation; and keep and real. In such a sense,

Which satisfies the equation,

Or fiddle with Tommy's gaussian approach. Much of this paper extends to all ; I just kept it with to keep it simpler. Theoretically the beta method works for all . Not too sure about complex values yet.

Regards, James