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 Could be tetration if this integral converges mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 05/03/2014, 01:19 AM (This post was last modified: 05/03/2014, 04:45 AM by mike3.) JmsNxn, I am a little suspicious of this method. In particular, I'm not sure the integral $I = \int_{0}^{\infty} \vartheta(x) x^{z-1} dx$, $0 < z < 1$ converges. I have done some numerical tests and it seems that the sup of $|\vartheta(x)|$ for $0 < x < X$ grows to infinity as $X \rightarrow \infty$ and grows fast enough that the decay of $x^{z-1}$ does not suppress it. This is just an experimental result, I'm not sure of the formal proof of divergence yet. In addition to this, I am suspect of the method used to show uniqueness, in particular, the use of the 1-cyclic warping of the tetration. sheldonison had constructed here: http://math.eretrandre.org/tetrationforu...p?pid=5019 an alternate tetration function which decays to a different, non-principal set of fixed points of the logarithm at $\pm i \infty$. Such a function (or its reciprocal $\frac{1}{\mathrm{tet}_{\mathrm{alt}}(-z)}$) has the asymptotic behavior you want, yet is different than the "usual" tetrational. That is, both $\mathrm{tet}$ and the alternate $\mathrm{tet}_{\mathrm{alt}}$ constructed have the property that their mirrored reciprocals $\frac{1}{F(-z)}$ satisfy $|\frac{1}{F(-z)}| < Ce^{a|\Im(z)|}$ for $0 < a < \pi/2$ in a vertical strip with $\Re(z) < 1$ which does not include points arbitrarily close to $1$ since they are bounded in such a strip. I suspect these are related by a 1-cyclic mapping which is such that it is not entire but instead has branch singularities, and so the warping requires a more careful treatment. In particular, you can get the warping by restricting to the real axis, then applying the 1-cyclic map, then analytically continuing again to the plane. The branch nature precludes a simple substitution on the whole plane. Also, are you sure you have that right, that a tetration function $F(z)$ should satisfy $|\frac{1}{F(-z)}| < Ce^{a|\Im(z)| + \rho|\Re(z)|}$ for some $0 < a < \pi/2$ and $\rho \ge 0$ with $\Re(z) < 1$ ? I believe that any tetration function $F(z)$ which is holomorphic for $\Re(z) \ge 0$ must take on values arbitrarily close to $0$, so that its reciprocal is unbounded, and thus cannot satisfy the exponential bound on an entire half-plane (but it can on a strip, of course). This can be shown from the chaos of the exponential map $e^z$. A tetration function (more correctly, a superfunction of the exponential function) satisfies $F(z+1) = e^{F(z)}$. The exponential map is topologically transitive, which means that if we have an open set $A$, and another $B$, we can find an integer n such that $\exp^n(A)$ contains at least one point of $B$. In particular, if we let $A = \{ F(z) : z \in \mathbb{C}\ \mathrm{and}\ 0 < \Re(z) < 1 \}$, which is open by the openness of the strip and the open mapping theorem, we have for any open $\epsilon$-disc $D(\epsilon)$ around $0$, no matter how small $\epsilon$, there is an n such that $\exp^n(A)$, and thus $F$ of the strip $n < \Re(z) < n + 1$ contains a point in that disc, hence within $\epsilon$ of 0, and so the reciprocal $\frac{1}{F(z)}$ must be unbounded on $\Re(z) > 0$, thus $\frac{1}{F(-z)}$ on $\Re(z) < 0$ and $\Re(z) < 1$ is also so unbounded. « Next Oldest | Next Newest »

 Messages In This Thread Could be tetration if this integral converges - by JmsNxn - 04/03/2014, 02:14 PM RE: Could be tetration if this integral converges - by sheldonison - 04/30/2014, 11:17 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/02/2014, 03:33 PM RE: Could be tetration if this integral converges - by tommy1729 - 04/30/2014, 12:29 PM RE: Could be tetration if this integral converges - by mike3 - 05/03/2014, 01:19 AM RE: Could be tetration if this integral converges - by tommy1729 - 05/11/2014, 04:26 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/11/2014, 04:30 PM RE: Could be tetration if this integral converges - by tommy1729 - 05/11/2014, 04:52 PM RE: Could be tetration if this integral converges - by mike3 - 05/03/2014, 05:24 AM RE: Could be tetration if this integral converges - by mike3 - 05/03/2014, 07:13 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/03/2014, 06:12 PM RE: Could be tetration if this integral converges - by mike3 - 05/04/2014, 02:18 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/04/2014, 07:27 PM RE: Could be tetration if this integral converges - by mike3 - 05/05/2014, 12:55 AM RE: Could be tetration if this integral converges - by mike3 - 05/04/2014, 11:50 AM RE: Could be tetration if this integral converges - by sheldonison - 05/04/2014, 03:28 PM RE: Could be tetration if this integral converges - by mike3 - 05/05/2014, 01:00 AM RE: Could be tetration if this integral converges - by sheldonison - 05/05/2014, 03:49 PM RE: Could be tetration if this integral converges - by tommy1729 - 05/04/2014, 01:25 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/04/2014, 07:36 PM RE: Could be tetration if this integral converges - by MphLee - 05/04/2014, 07:44 PM RE: Could be tetration if this integral converges - by mike3 - 05/04/2014, 10:42 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/04/2014, 11:32 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/04/2014, 09:06 PM RE: Could be tetration if this integral converges - by mike3 - 05/05/2014, 02:11 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/05/2014, 04:27 PM RE: Could be tetration if this integral converges - by mike3 - 05/05/2014, 11:45 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/06/2014, 12:11 AM RE: Could be tetration if this integral converges - by mike3 - 05/06/2014, 06:50 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/06/2014, 03:54 PM RE: Could be tetration if this integral converges - by mike3 - 05/07/2014, 03:25 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/07/2014, 03:18 PM RE: Could be tetration if this integral converges - by mike3 - 05/11/2014, 07:47 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/11/2014, 04:29 PM RE: Could be tetration if this integral converges - by mike3 - 05/11/2014, 11:26 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/12/2014, 01:44 AM RE: Could be tetration if this integral converges - by mike3 - 05/12/2014, 02:15 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/12/2014, 03:32 PM RE: Could be tetration if this integral converges - by tommy1729 - 05/12/2014, 11:26 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/13/2014, 01:58 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/05/2014, 06:18 PM RE: Could be tetration if this integral converges - by tommy1729 - 05/05/2014, 09:09 PM

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