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 Searching for an asymptotic to exp[0.5] sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 05/10/2014, 12:14 PM (This post was last modified: 05/10/2014, 07:41 PM by sheldonison.) (05/08/2014, 04:25 PM)sheldonison Wrote: (05/07/2014, 12:22 PM)tommy1729 Wrote: Im searching for an asymptotic to exp[0.5](x)....Emperical testing suggests that an "entire" pseudo half iterate is very likely possible, with all positive Taylor series coefficients at z=0, and a probable growth value of 0.5, as defined by the "growth" equation. .... Each derivative is bounded to a maximum value by a particular value of half(z)....still working on how to formalize the definition of the conjectured Taylor series. - Sheldon $\text{halfassym}(z) = \sum_{n = 1}^{\infty}\frac{x^n}{a_n!}$ Here, for a_n, factorial is extended to the reals with the gamma(n+1) function. I do not know what the limiting equation for a_n is as n gets arbitrarily large. These values were generated numerically. With 150 Taylor series terms, this series can accurately generate the half iterate for numbers up to 10^12. Code:a_1= 1.289368074687 a_2= 2.685542084449 a_3= 4.481892104368 a_4= 6.478743767633 a_5= 8.617330003873 a_6= 10.86883764614 a_7= 13.21549097355 a_8= 15.64480318533 a_9= 18.14765340676 a_10= 20.71664375389 a_11= 23.34605172725 a_12= 26.03107295368 a_13= 28.76759185629 a_14= 31.55218923626 a_15= 34.38183207053 a_16= 37.25395409452 a_17= 40.16617725981 a_18= 43.11650203072 a_19= 46.10306787913 a_20= 49.12423329725 a_21= 52.17844415943 a_22= 55.26438661508 a_23= 58.38072977033 a_24= 61.52640919181 a_25= 64.70028943211 a_26= 67.90141159249 a_27= 71.12888751647 a_28= 74.38183008946 a_29= 77.65944293902 a_30= 80.96100674887 a_31= 84.28582227410 a_32= 87.63323120070 a_33= 91.00261369743 a_34= 94.39338557884 a_35= 97.80499516680 a_36= 101.2369199152 a_37= 104.6886902658 a_38= 108.1598101190 a_39= 111.6498571303 a_40= 115.1583810162 a_41= 118.6850011605 a_42= 122.2293283987 a_43= 125.7909699804 a_44= 129.3696195111 a_45= 132.9648897794 a_46= 136.5764573911 a_47= 140.2040557334 a_48= 143.8473645074 a_49= 147.5060942647 a_50= 151.1799685187 a_51= 154.8686996662 a_52= 158.5720762206 a_53= 162.2898267064 a_54= 166.0217152588 a_55= 169.7675306959 a_56= 173.5270093037 a_57= 177.2999786775 a_58= 181.0862092231 a_59= 184.8855000278 a_60= 188.6976766463 a_61= 192.5225306431 a_62= 196.3598778123 a_63= 200.2095425885 a_64= 204.0713710077 a_65= 207.9451779799 a_66= 211.8308191885 a_67= 215.7281180480 a_68= 219.6369408364 a_69= 223.5570997849 a_70= 227.4885071866 a_71= 231.4309843808 a_72= 235.3843931661 a_73= 239.3486192792 a_74= 243.3235140576 a_75= 247.3089677325 a_76= 251.3048375095 a_77= 255.3110228135 a_78= 259.3273875110 a_79= 263.3538324564 a_80= 267.3902471867 a_81= 271.4365024525 a_82= 275.4924928105 a_83= 279.5581471603 a_84= 283.6333119804 a_85= 287.7179392030 a_86= 291.8118947335 a_87= 295.9151023881 a_88= 300.0274504396 a_89= 304.1488634944 a_90= 308.2792519888 a_91= 312.4185117624 a_92= 316.5665730911 a_93= 320.7233354589 a_94= 324.8887317451 a_95= 329.0626821729 a_96= 333.2451078801 a_97= 337.4359118041 a_98= 341.6350372412 a_99= 345.8424068745 a_100= 350.0579336860 a_101= 354.2815588969 a_102= 358.5132133215 a_103= 362.7528106159 a_104= 367.0003160828 a_105= 371.2556293444 a_106= 375.5186854508 a_107= 379.7894500858 a_108= 384.0678314842 a_109= 388.3537801607 a_110= 392.6472189586 a_111= 396.9481223641 a_112= 401.2563957391 a_113= 405.5719986392 a_114= 409.8948585083 a_115= 414.2249480026 a_116= 418.5621839981 a_117= 422.9065248225 a_118= 427.2579174535 a_119= 431.6162930147 a_120= 435.9816335167 a_121= 440.3538537394 a_122= 444.7329047543 a_123= 449.1187668605 a_124= 453.5113755327 a_125= 457.9106702550 a_126= 462.3166165089 a_127= 466.7291677505 a_128= 471.1482630381 a_129= 475.5738880517 a_130= 480.0059669786 a_131= 484.4444725900 a_132= 488.8893596420 a_133= 493.3405854525 a_134= 497.7981092466 a_135= 502.2618743290 a_136= 506.7318543022 a_137= 511.2080081242 a_138= 515.6902967113 a_139= 520.1786803772 a_140= 524.6731047065 a_141= 529.1735644829 a_142= 533.6799885996 a_143= 538.1923555703 a_144= 542.7106288953 a_145= 547.2347567376 a_146= 551.7647258028 a_147= 556.3004650155 a_148= 560.8419393211 a_149= 565.3890426629 a_150= 569.9416519995 One can compare this to Tommy's hypothetical candidate (quote below), and see that the (n^2)! in the denominator is growing much quicker than necessary, as compared to the empirical results. But one can also see that the denominator in the Taylor series coefficients for this asymptotic half iterate grow much faster than for the exp(z). The conjecture is as n gets arbitrarily large, for any z0>1, the slog(f^n(z0))/n~=0.5, and therefore this would be an entire function with half exponential growth. tommy1729 Wrote:f1(z) = sum z^n/(n^2)! The asymptotic half iterate function is defined such that all Taylor series coefficients are positive, and that the function is always less than but approaching the Kneser half iterate, for real(z)>0. The construction for the Taylor series I used is a somewhat complicated two stage process; I'll post more later. But the first stage is to note that if the Taylor series terms are all positive, than no individual Taylor series term can be larger than the desired sum, so we require that, $\forall {x>0} \; a_n x^n \lt \text{half}(x)$. This gives an upper bounds for the Taylor series coefficients. This function is always bigger than the Kneser half(z) function. The second stage is to scale down the terms by observing that for each value of half(z), one particular Taylor series term is largest contributer to the Taylor series sum. That determines for each term how much to scale that term down by. I think the resulting equation will always be less than the Kneser half iterate, but the ratio of this function over the Kneser half iterate will approach arbitrarily close to 1. Because the function is bounded to the right by the Kneser half iterate, we can safely say that this assymptotic half iterate is entire, assuming the construction works for arbitrarily large values of z and arbitrarily large Taylor series terms. Here are some example of calculations using this assymptotic half Taylor series, as compared to the Kneser half iterate. It would probably make sense to set a_0 of the half iterate to sexp(-0.5), which is the half iterate of 0. I will half to generate a complex plan plot for this half iterate... z, assymptotic_half, Kneser_half 0 0 0.4985632879411 1 1.126644950749 1.646354233751 10 58.93202104249 61.48617436731 100 187646.5930113 192708.5721853 1000 425414280682.2 432750850493.0 10000 9.638915213265 E21 9.750966938073 E21 100000 5.362748331798 E37 5.406412389290 E37 1000000 3.362567348729 E60 3.382539228002 E60 10000000 2.187210706560 E92 2.196854946875 E92 100000000 2.935957885769 E135 2.945782901678 E135 1000000000 3.788233214763 E192 3.798003781412 E192 10000000000 5.577154174589 E266 5.588492690694 E266 100000000000 3.101249943705 E361 3.106297055696 E361 1000000000000 6.614359301415 E480 6.622925643007 E480 One obvious questions from the Taylor series result, that I can't answer, because I have no idea how fast these functions grow as x goes to infinity, relative to exponentiaton. What is the "growth" of a functions like these, which should grow slower than exponentiation, but faster than any polynomial? $\sum_{n = 1}^{\infty}\frac{x^n}{(2n)!}$ $\sum_{n = 1}^{\infty}\frac{x^n}{(4n)!}$ « 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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