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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, 05:24 AM (This post was last modified: 05/03/2014, 01:14 PM by mike3.) On the other hand, what I saw here: http://en.wikipedia.org/wiki/Mellin_inversion_theorem suggests that the Mellin inverse transform requires only the right boundedness in vertical strips, not on a whole half-plane. So it should work... ... but then I tried a numerical test to compare the value of $\vartheta(w)$ against the inverse Mellin transform of $\frac{\Gamma(s)}{^{-s} e}$ using the Kneser tetrational function as obtained from sheldonison's excellent Kneser-method code (which satisfies the required boundedness criteria). The approximation to the inverse Mellin transform is done by numerically integrating $IM(x) \approx \frac{1}{2\pi i} \int_{c - iA}^{c + iA} x^{-s} \frac{\Gamma(s)}{^{-s} e} ds$ with $A$ some large real value (here, I chose 15) and $c$ is any number in $(0, 1)$ (here, I chose 0.5). The following graph shows $IM(x)$ and $\vartheta(x)$ on the positive reals from near-0 to 20:     As you can see, $IM(x)$ decays toward 0, but $\vartheta(x)$ does not. It seems that $IM(x) \approx \vartheta(x)$ for small $x$, but the approximation gets worse at larger values of $x$. This suggests that there is an error in your derivation of $\vartheta(x)$ from the inverse Mellin transform, since you should be able to Mellin-transform it back, which means your $\vartheta(x)$ should decay to 0 along the real axis, but it doesn't. I suspect a formal disproof could be had if you can show that $\mathrm{lim\ sup}_{x \rightarrow \infty} \ \vartheta(x) x^{s-1} = +\infty$. Whatever your $\vartheta(x)$ is, it seems most assuredly not to be the inverse Mellin transform of $\frac{\Gamma(s)}{^{-s} e}$, at least not for the Kneser tetrational (and considering it doesn't seem to provide a convergent Mellin transform, perhaps not the inverse Mellin of anything). « 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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