• 1 Vote(s) - 5 Average
• 1
• 2
• 3
• 4
• 5
 Could be tetration if this integral converges mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 05/11/2014, 11:26 PM (This post was last modified: 05/11/2014, 11:29 PM by mike3.) (05/11/2014, 04:29 PM)JmsNxn Wrote: If $F$ is holomorphic for $\Re(z) < b$ and $|F(z)| < C e^{\alpha |\im(z)| + \rho|\Re(z)|}$ then: $\beta(x) = \sum_{n=0}^\infty \frac{(-x)^n}{n!}F(-n)$ When F has singularities we see we pull on a second balancing function $\psi$ such that: $\beta(x) = \psi(x) + \sum_{n=0}^\infty \frac{(-x)^n}{n!}F(-n)$ For simple functions like $F(z) = 1/(z^2 - 1)$ then $\psi$ is very easy to calculate. For more complicated functions like tetration, which no doubt has essential singularities instead of poles, it becomes much more complicated. But the general result on essential singularities is just applying cauchy's residue formula around all the poles of $F$ on the function $\G(z)x^{-z}$ Well, I guess then these formulas aren't of much use for continuum-summing tetration, since it is most definitely not bounded with the bound $|F(z)| < C e^{\alpha |\Im(z)| + \rho |\Re(z)|}$ . In fact it is unbounded on the right half-plane (where it behaves chaotically) and has branch point singularities (which are neither poles nor essential singularities) on the left half-plane (these are logarithmic, double-logarithmic, triple-logarithmic, etc. in that order at $z = -2$, $z = -3$, $z = -4$, ...). I'm curious: how did you get that first formula? Is it possible to get a similar formula for $\mathrm{semi}\beta_U(x) = \frac{1}{2\pi i} \int_{\sigma}^{\sigma + i\infty} \Gamma(z) F(z) x^{-z} dz$ and $\mathrm{semi}\beta_L(x) = \frac{1}{2\pi i} \int_{\sigma - i\infty}^{\sigma} \Gamma(z) F(z) x^{-z} dz$ and $F$ satisfying the given bound? As then I might have something, perhaps. I'll have to see, though. « 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

 Possibly Related Threads... Thread Author Replies Views Last Post Where is the proof of a generalized integral for integer heights? Chenjesu 2 3,472 03/03/2019, 08:55 AM Last Post: Chenjesu [integral] How to integrate a fourier series ? tommy1729 1 4,286 05/04/2014, 03:19 PM Last Post: tommy1729 Some integral transforms related to tetration JmsNxn 0 2,939 05/02/2013, 07:54 PM Last Post: JmsNxn (draft) integral idea tommy1729 0 3,448 06/25/2011, 10:17 PM Last Post: tommy1729 Cauchy integral also for b< e^(1/e)? bo198214 14 20,933 04/24/2009, 05:29 PM Last Post: bo198214

Users browsing this thread: 1 Guest(s)