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 Using a family of asymptotic tetration functions... JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 08/06/2021, 01:47 AM (This post was last modified: 08/07/2021, 04:15 AM by JmsNxn.) (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 $f(z+1)=a^z*b^f(z)$ with arbitrary constant a and b I simply take $f(z)=g(a^z)$ and solve $g(az)=z*b^g(z)$ term by term in series, which uses this code I wrote by wolfram mathematica 12, it iterates the recurrence to get the function converged 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] < 1I 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: $\lim_{a\to1}f(z-g(a))\sim\mathrm{tet}_b(z)$ 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 $b^f(z)$, particularly? Regards, James OHHHH WWAIT, nevermind, I get it. You meant to write $b^{f(z)}$.  You are absolutely correct. What you have constructed here; using Sheldon's idea of a modified Schroder function; you've made, $ f(s) = \Omega_{j=1}^\infty a^{s-j} b^z\,\bullet z\\$ This function will be holomorphic for $|a| > 1 ,b \neq 0,s \in \mathbb{C}$. This is similar to how I constructed the $\phi$ method, where I took $a = b = e$. The conjecture that stands is that this can only construct a $\mathcal{C}^\infty$ tetration on $\mathbb{R}^+$. And converges nowhere in $\mathbb{C}$ 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, $ F(a,b,s) = \lim_{n\to\infty} \log^{\circ n} f(s+n)\\ \text{for}\,\,a>1\,b > 0\,s \in \mathbb{R}\,s > R\,\,\text{for some}\,\,R > 0\\ b^{F(a,b,s)} = F(a,b,s+1)\\$ It will probably diverge in $\mathbb{C}$ though. It's going to look like the $\phi$ method. I'd suggest looking at something that solves the asymptotic equation; and keep $b > e^{1/e}$ and real.  In such a sense, $ g(b,\lambda, s) = \Omega_{j=1}^\infty \frac{b^z}{e^{\lambda(j-s)} + 1}\,\bullet z\\$ Which satisfies the equation, $ \log_b g(b,\lambda, s+1) = g(b,\lambda,s) - \log_b(1+e^{-\lambda s})\\$ Or fiddle with Tommy's gaussian approach. Much of this paper extends to all $b > e^{1/e}$; I just kept it with $e$ to keep it simpler. Theoretically the beta method works for all $b > e^{1/e}$. Not too sure about complex values yet. Regards, James « Next Oldest | Next Newest »

 Messages In This Thread Using a family of asymptotic tetration functions... - by JmsNxn - 04/01/2021, 05:19 AM RE: Using a family of asymptotic tetration functions... - by JmsNxn - 04/05/2021, 08:43 AM RE: Using a family of asymptotic tetration functions... - by JmsNxn - 04/11/2021, 01:01 AM RE: Using a family of asymptotic tetration functions... - by JmsNxn - 04/23/2021, 04:26 AM RE: Using a family of asymptotic tetration functions... - by JmsNxn - 04/24/2021, 08:26 AM RE: Using a family of asymptotic tetration functions... - by JmsNxn - 04/24/2021, 08:54 PM RE: Using a family of asymptotic tetration functions... - by MphLee - 04/24/2021, 11:09 PM RE: Using a family of asymptotic tetration functions... - by JmsNxn - 04/26/2021, 01:28 AM RE: Using a family of asymptotic tetration functions... - by sheldonison - 04/26/2021, 02:11 AM RE: Using a family of asymptotic tetration functions... - by JmsNxn - 04/27/2021, 04:28 AM RE: Using a family of asymptotic tetration functions... - by MphLee - 04/26/2021, 10:24 AM RE: Using a family of asymptotic tetration functions... - by Ember Edison - 05/03/2021, 01:58 PM RE: Using a family of asymptotic tetration functions... - by JmsNxn - 05/03/2021, 08:06 PM RE: Using a family of asymptotic tetration functions... - by JmsNxn - 05/10/2021, 02:26 AM RE: Using a family of asymptotic tetration functions... - by Leo.W - 08/05/2021, 04:51 PM RE: Using a family of asymptotic tetration functions... - by JmsNxn - 08/06/2021, 01:47 AM

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