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 The upper superexponential Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 05/12/2009, 08:54 AM (05/11/2009, 12:55 PM)sheldonison Wrote: Are $F_{2,3}$ and $F_{4,3}$ the same two functions in the Bummer post?Yes. These are the same functions. Henryk asked me to plot them all versus real argument as a separate post. (05/11/2009, 12:55 PM)sheldonison Wrote: Could you comment on how the behavior of the two functions differ in the complex plane?Yes. Functions $F_{4,5}$ and $F_{4,5}$ entire. One of them can be obtained from another one, just displacing the argument. Tetration $F_{2,1}$ has, as you know, singularities and the cutline; due to the periodicity, there is set of singulatities and cutlines. Function $F_{2,3}$ can be obtained by translation of tetration $F_{2,1}$, so, it has similar singulatities. [/quote] (05/11/2009, 12:55 PM)sheldonison Wrote: Do both functions have the same values at z=+/-i*infinity?No. There is no need to talk about values $\pm \infty$, because each of them is periodic and the periods are imaginary. (05/11/2009, 12:55 PM)sheldonison Wrote: Do they have the same periodicity?No. Periods are different: $T_2=2\pi \mathrm{i}/\ln(\ln(2))\approx -17.143148179354847104 {\mathrm i}$ $T_4=2\pi \mathrm{i}/\ln(2\ln(2))\approx 19.236149042042854712 {\mathrm i}$ (05/11/2009, 12:55 PM)sheldonison Wrote: Does only one have singularities?Tetration $F_{2,1}$ has singularities; its displacement $F_{2,3}$ has too. (05/11/2009, 12:55 PM)sheldonison Wrote: Given that $F_{2,3}(z)= F_{4,3}(z)$ at all integer values of z, then can these two functions be expressed in terms of each other, where $F_{2,3}= F_{4,3}(x+\theta(x))$?Yes. (05/11/2009, 12:55 PM)sheldonison Wrote: Is the $\theta(x)$ function analytic?Yes. $\theta(z)=F_{4,3}^{-1}(F_{2,3}(z)) -z$ is 1-periodic function; it is almost sinusoidal. Henryk, can we begin to distribute the draft of our paper? It would answer a lot of questions we provoked with the plot... « Next Oldest | Next Newest »

 Messages In This Thread The upper superexponential - by bo198214 - 03/29/2009, 11:23 AM RE: The upper superexponential - by andydude - 03/31/2009, 04:29 AM RE: The upper superexponential - by sheldonison - 04/03/2009, 03:06 PM RE: The upper superexponential - by bo198214 - 04/03/2009, 04:22 PM RE: The upper superexponential - by sheldonison - 04/05/2009, 12:45 PM RE: The upper superexponential - by bo198214 - 04/06/2009, 06:35 AM RE: The upper superexponential - by Kouznetsov - 05/10/2009, 02:13 PM RE: The upper superexponential - by sheldonison - 05/11/2009, 12:55 PM RE: The upper superexponential - by bo198214 - 05/11/2009, 01:21 PM RE: The upper superexponential - by sheldonison - 05/11/2009, 08:12 PM RE: The upper superexponential - by bo198214 - 05/11/2009, 08:31 PM RE: The upper superexponential - by Kouznetsov - 05/12/2009, 08:54 AM RE: The upper superexponential - by bo198214 - 06/01/2009, 07:24 PM RE: The upper superexponential - by tommy1729 - 04/05/2009, 07:05 PM RE: The upper superexponential - by sheldonison - 04/22/2009, 05:02 PM RE: The upper superexponential - by bo198214 - 04/22/2009, 05:34 PM RE: The upper superexponential/eich-value x for h=0? - by Gottfried - 09/18/2009, 04:01 PM

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