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 Searching for an asymptotic to exp[0.5] sheldonison Long Time Fellow Posts: 640 Threads: 22 Joined: Oct 2008 09/03/2014, 01:04 PM (This post was last modified: 09/03/2014, 01:31 PM by sheldonison.) (09/01/2014, 10:24 PM)tommy1729 Wrote: It seems my friend ( our ? ) mick has given us ( myself and sheldon in particular ) credit for the use of " fake function theory " here is the link : http://math.stackexchange.com/questions/...for-real-x As I mentioned before z - ln(2sinh(z)/z) is asymptotic to ln(z) near the real line, but it fails to be entire ( sinh is periodic ! ). Guess we could consider the fake ln(x^2 + 1) here too. Unlike the half-exp we are considering more standard functions which might give intresting closed form results !... Thanks for the link Tommy! I've been overseas vacationing... I was able to use the basic recipe from this post to interpolate $\ln(x^2+1)\exp(x^2)$ as an even function, just as you suggested! I will post more later, either here or at Mathstack. I have also been working on an example that should allow us to make the "fake function" theory more rigorous. A very interesting asymptotic function is below. It is interesting because the corresponding $\ln(f(\exp(z)))$ function is simply $\frac{x^2}{2}$, whose derivative is x, so $h(n)=n$, so all of the Taylor series coefficients of the interpolating function have an exact closed form, as do all of the error equations. $f(x) = \exp \left$$\frac{(\ln(x))^2}{2} \right$$$ $a_n = \frac{1}{\exp(0.5n^2)\sqrt{2\pi}}\;\;$ for this function, the Gaussian approximation is also the best approximation. $f(x) \approx \sum_{n=-\infty}^{\infty} a_n x^n\;\;$ this is a converging Laurent series. We don't use the negative coefficients for the entire interpolant. The Laurent series interpolant can also be expressed in a quickly converging closed form. $\sum_{n=-\infty}^{\infty} a_n x^n\;=\;\sum_{n=-\infty}^{\infty} \exp \left$$\frac{(\ln(x)+2n\pi i)^2}{2} \right$$\;\;$ f(x) corresponds to n=0 - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/07/2014, 12:22 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/08/2014, 04:25 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/10/2014, 12:14 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/10/2014, 11:31 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/10/2014, 11:48 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/10/2014, 11:58 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/09/2014, 11:19 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/10/2014, 11:56 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/13/2014, 04:23 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/14/2014, 05:54 AM RE: Searching for an asymptotic to exp[0.5] - by JmsNxn - 05/12/2014, 03:48 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/12/2014, 03:56 PM RE: Searching for an asymptotic to exp[0.5] - by JmsNxn - 05/12/2014, 05:06 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/12/2014, 11:35 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/13/2014, 11:44 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/14/2014, 11:42 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/15/2014, 06:15 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/15/2014, 09:49 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/16/2014, 07:27 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/14/2014, 12:20 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 07/17/2014, 05:46 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/15/2014, 08:53 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/16/2014, 09:36 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/16/2014, 10:16 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/18/2014, 06:14 PM RE: Searching for an asymptotic to exp[0.5] - by JmsNxn - 05/22/2014, 12:16 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/22/2014, 07:08 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/22/2014, 08:31 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/22/2014, 10:16 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/23/2014, 10:53 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/25/2014, 03:00 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/29/2014, 11:09 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 07/12/2014, 07:46 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/29/2014, 11:32 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/29/2014, 11:54 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/30/2014, 09:41 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 06/28/2014, 11:16 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/28/2014, 09:52 PM RE: Searching for an asymptotic to exp[0.5] - by JmsNxn - 06/29/2014, 01:40 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 06/30/2014, 12:56 AM RE: Searching for an asymptotic to exp[0.5] - by JmsNxn - 06/30/2014, 03:21 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 06/30/2014, 11:56 PM RE: Searching for an asymptotic to exp[0.5] - by JmsNxn - 07/01/2014, 12:35 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 06/30/2014, 01:27 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/01/2014, 10:10 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 07/01/2014, 11:41 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/02/2014, 09:53 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/10/2014, 11:48 PM RE: Searching for an asymptotic to exp[0.5] - by MorgothV8 - 07/13/2014, 06:48 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/14/2014, 12:27 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/14/2014, 11:16 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/15/2014, 08:22 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/15/2014, 09:43 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/15/2014, 09:48 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/24/2014, 12:10 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/24/2014, 10:47 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 07/25/2014, 02:46 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/24/2014, 10:54 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/26/2014, 12:21 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/27/2014, 08:37 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/27/2014, 05:01 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/28/2014, 12:17 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/28/2014, 10:30 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 07/30/2014, 04:07 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/01/2014, 11:20 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/01/2014, 11:36 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/02/2014, 12:26 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 08/02/2014, 03:44 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/02/2014, 11:02 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/02/2014, 11:48 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 08/03/2014, 04:54 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/03/2014, 08:46 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 08/03/2014, 12:06 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/03/2014, 12:10 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/05/2014, 11:31 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/08/2014, 10:28 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/09/2014, 12:24 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 08/10/2014, 06:08 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/01/2014, 10:24 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 09/03/2014, 01:04 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/02/2014, 07:46 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/02/2014, 07:53 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/08/2014, 12:56 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 09/08/2014, 04:15 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/08/2014, 11:03 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 09/09/2014, 04:33 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 09/09/2014, 06:26 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/10/2014, 11:02 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/11/2014, 08:02 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 09/11/2014, 02:13 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/12/2014, 07:49 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/12/2014, 06:35 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 09/13/2014, 07:15 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/13/2014, 11:25 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 09/13/2014, 11:45 PM RE: Searching for an asymptotic to exp[0.5] - by jaydfox - 09/13/2014, 11:49 PM RE: Searching for an asymptotic to exp[0.5] - by jaydfox - 09/14/2014, 12:00 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/14/2014, 05:07 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 09/15/2014, 03:53 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/14/2014, 09:34 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/16/2014, 12:14 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/16/2014, 12:27 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/18/2014, 10:20 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/18/2014, 11:07 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/19/2014, 12:23 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/29/2014, 11:40 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/19/2014, 04:02 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 11/03/2014, 01:26 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 11/03/2014, 10:49 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 11/03/2014, 11:34 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 11/03/2014, 11:39 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 11/04/2014, 09:41 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 11/04/2014, 10:38 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 11/05/2014, 11:58 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 11/07/2014, 12:27 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 03/28/2015, 11:11 PM RE: Searching for an asymptotic to exp[0.5] - by marraco - 03/29/2015, 12:59 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/25/2015, 10:24 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 05/25/2015, 10:52 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/15/2015, 06:45 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/15/2015, 06:55 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/17/2015, 01:45 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/18/2015, 09:34 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/18/2015, 09:56 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/18/2015, 10:09 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/31/2015, 04:57 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/31/2015, 05:12 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/15/2015, 10:22 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/16/2015, 02:49 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/16/2015, 03:23 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 08/26/2015, 07:36 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/03/2015, 10:31 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/05/2015, 08:16 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/09/2015, 12:17 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/12/2015, 01:14 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/14/2015, 01:30 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/18/2015, 11:31 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/21/2015, 10:53 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 09/21/2015, 05:58 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/24/2015, 08:10 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/25/2015, 12:59 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 09/25/2015, 08:26 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/26/2015, 12:24 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/29/2015, 12:28 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 10/01/2015, 07:56 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/30/2015, 12:25 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/30/2015, 09:27 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/01/2015, 11:25 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/02/2015, 02:56 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/03/2015, 10:42 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/06/2015, 12:11 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/06/2015, 12:26 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/08/2015, 07:52 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/08/2015, 12:26 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/08/2015, 10:43 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/08/2015, 11:08 PM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 10/09/2015, 08:15 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/09/2015, 11:56 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 10/10/2015, 03:08 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/10/2015, 07:40 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/08/2015, 11:12 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/09/2015, 07:18 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/10/2015, 08:15 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/10/2015, 08:26 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/11/2015, 07:17 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/17/2015, 11:59 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/18/2015, 11:07 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/18/2015, 11:22 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/19/2015, 12:20 AM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 10/27/2015, 01:27 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 02/16/2016, 03:17 AM RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 02/18/2016, 06:51 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 02/17/2016, 01:25 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 02/18/2016, 12:53 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 02/18/2016, 01:11 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 02/23/2016, 01:01 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 02/23/2016, 01:23 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 03/21/2016, 01:26 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 04/05/2016, 01:29 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/04/2016, 07:21 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/04/2016, 08:16 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/04/2016, 08:29 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/06/2016, 03:12 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/06/2016, 03:47 PM RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 03/15/2018, 01:23 PM

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