entire function close to sexp ?? - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: entire function close to sexp ?? (/showthread.php?tid=851) entire function close to sexp ?? - tommy1729 - 04/26/2014 Is there an entire function known that is close to sexp(z) near the positive real line ? Such a function must exist. Never read about it though. Apart from interpolation I do not know in what way this should be constructed. regards tommy1729 RE: entire function close to sexp ?? - tommy1729 - 04/27/2014 This smells like an integral transform. And a proof with contour integration. Exactly how is another thing. Maybe something for James Nixon ? (user JmsNxn's name I assume) regards tommy1729 RE: entire function close to sexp ?? - JmsNxn - 04/29/2014 I got a function that is very close to tetration: Let us take the function: $\vartheta(w) = \sum_{n=0}^\infty \frac{w^n}{n!(^ne)}$ Then $\vartheta$ is exponential order zero. If $\int_0^\infty |\vartheta(-w)|w^{\sigma-1}\,dw<\infty$ for $0 < \sigma < 1$ Then $\int_0^\infty \vartheta(-w)w^{s-1} \,dw = \frac{\Gamma(s)}{(^{-s} e)}$ Which IS TETRATION and satisfies the recursion. However, I haven't been able to prove absolute convergence of that integral. BUT! We can take: $\phi(s) = \frac{1}{\Gamma(s)}\int_0^\infty e^{-\lambda w}\vartheta(-w)w^{s-1}\,dw$ for $0<\lambda<\epsilon$ and $\phi(s) \approx \frac{1}{(^{-s}e)}$ for small $\epsilon$ hopefully not too small to blow up. Furthermore this can be made even better for $\forall s \in \mathbb{C}$ $\frac{1}{(^s e)} \approx \psi(s) = \frac{1}{\Gamma(-s)} (\sum_{n=0}^\infty \frac{(-1)^n}{n!(^n e)(n-s)} + \int_1^\infty e^{-\lambda w}\vartheta(-w)w^{-s-1}\,dw)$ These functions are so close to tetration it makes me want to punch a hole in the wall that I can' prove absolute convergence. -_- lol. It's Real to real and interpolates tetration too. The limit as $\lambda \to 0$ is tetration in both cases if it converges. but both functions are not the same $\psi \neq \phi$. RE: entire function close to sexp ?? - tommy1729 - 04/29/2014 Im a bit puzzled by the " -s ". e^[3] = e^(e^e) But e^[-3] = ln(ln(ln(1))) = ln(ln(0)) = ?? So I do not know what the "-s" means. However your last formula seems to end that problem so it seems $\frac{1}{(^s e)} \approx \psi(s) = \frac{1}{\Gamma(-s)} (\sum_{n=0}^\infty \frac{(-1)^n}{n!(^n e)(n-s)} + \int_1^\infty e^{-\lambda w}\vartheta(-w)w^{-s-1}\,dw)$ Is the kind of formula we look for. However, Im more concerned about wheither or not this is analytic !? Convergeance is an intresting subject, but if it is analytic somewhere then by analytic continuation we can extend it. But why should that be analytic ? Assuming it is analytic then its derivative must equal the term by term derivative. Can you show it - the term by term derivative formulation - converges ? Actually I bet that this function is not analytic and I even bet sheldon agrees with me. Intresting though. But I suspect you have mentioned this function before, not ? regards tommy1729 RE: entire function close to sexp ?? - JmsNxn - 04/29/2014 (04/29/2014, 09:25 PM)tommy1729 Wrote: Im a bit puzzled by the " -s ". e^[3] = e^(e^e) But e^[-3] = ln(ln(ln(1))) = ln(ln(0)) = ?? So I do not know what the "-s" means. However your last formula seems to end that problem so it seems $\frac{1}{(^s e)} \approx \psi(s) = \frac{1}{\Gamma(-s)} (\sum_{n=0}^\infty \frac{(-1)^n}{n!(^n e)(n-s)} + \int_1^\infty e^{-\lambda w}\vartheta(-w)w^{-s-1}\,dw)$ Is the kind of formula we look for. However, Im more concerned about wheither or not this is analytic !? Convergeance is an intresting subject, but if it is analytic somewhere then by analytic continuation we can extend it. But why should that be analytic ? Assuming it is analytic then its derivative must equal the term by term derivative. Can you show it - the term by term derivative formulation - converges ? Actually I bet that this function is not analytic and I even bet sheldon agrees with me. Intresting though. But I suspect you have mentioned this function before, not ? regards tommy1729 very easy to see this is analytic. No idea what you're talking about. lol. Mellin transform of an entire function is analytic. Multiply it by the inverse gamma which is entire and voila, analytic and entire. I'd explain why but it's literally an excercise in analysis 101. These are all uniform limits of analytic functions, so it's neessarily analytic. RE: entire function close to sexp ?? - sheldonison - 04/30/2014 (04/26/2014, 12:23 PM)tommy1729 Wrote: Is there an entire function known that is close to sexp(z) near the positive real line ? Such a function must exist. Never read about it though. Apart from interpolation I do not know in what way this should be constructed. regards tommy1729Tommy, There are many examples of entire superexponentials, such as iterating f(z)=b^z, where b=exp(1/e), jaydfox calls this "cheta(z)", fixpoint=e f(z)=exp(z)-1, this turns out to be cheta(z)/e-1, fixpoint=0 f(z)=2sinh(z), fixpoint=0 f(z)=sinh(z), fixpoint=0 f(z)=b^z, from the upper fixed point for bases less than 1