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 How it looks (i.θ)ₐ marraco Fellow Posts: 93 Threads: 11 Joined: Apr 2011 04/23/2015, 04:52 PM (This post was last modified: 04/24/2015, 05:08 AM by marraco.) (04/21/2015, 08:23 AM)sheldonison Wrote: and sexp(10) is a humongous number. The largest number pari-gp can represent is about $\text{sexp}_e\left(4.08\right)\approx 7.24676\cdot 10^{131475196{$That was my dumb mistake. I didn't needed sexp(10), but sexp(I*10). ¿Do you think that the tiny spirals at the end of the lines ͥ·ˣa (as x→∞) are accurate, or just a numerical artifact? ¿Did you attempted to find the base for which the lines turn into an "ellipse" (or something close)?. I guess a=e^(e^-1); that would "explain" the change of state. Now, if the bases$\vspace{15}{1really turn into ellipses, then it should be easy to find an algebraic expression for tetration to real exponents, or at least an important insight for $\vspace{20}{^{\frac{1}{x}}a}$ The base $\vspace{15}{a=e^{e^{-1}} }$ seems to match an ellipse with center near c=2.65599203615835 (Don't take that precision as accurate. I got it from Excel), relation of axis b=a, and radius $\vspace{20}{r=\left(e^{\pi}+W_{(1)}\right)^{\,e^{-\pi}}}$, or $\left(imag(^{i.x}a)\right)^2 \,+\, \left({\frac{real(^{i.x}a)-c}{b}}\right)^2\,=\, r^2\\ \\ a=e^{e^{-1}} \\ b= e^{e^{-1}}\\ c= 2.655992036 \\ r=\left(e^{\pi}+W_{(1)}\right)^{\,e^{-\pi}}\\ \\ W_{(x)}.e^{W_{(x)}}=x \,\Rightarrow \, e={\frac{1}{W_{(1)}}}^{\frac{1}{W_{(1)}}} \,\Rightarrow \, W_{(1)}=0,56714329 $ I have the result, but I do not yet know how to get it. « Next Oldest | Next Newest »

 Messages In This Thread How it looks (i.θ)ₐ - by marraco - 04/18/2015, 11:20 PM RE: How it looks (i.θ)ₐ - by sheldonison - 04/19/2015, 02:40 PM RE: How it looks (i.θ)ₐ - by marraco - 04/19/2015, 08:40 PM RE: How it looks (i.θ)ₐ - by sheldonison - 04/19/2015, 11:19 PM RE: How it looks (i.θ)ₐ - by marraco - 04/20/2015, 02:35 AM RE: How it looks (i.θ)ₐ - by Gottfried - 04/20/2015, 07:53 AM RE: How it looks (i.θ)ₐ - by marraco - 04/21/2015, 02:34 AM RE: How it looks (i.θ)ₐ - by marraco - 04/21/2015, 03:20 AM RE: How it looks (i.θ)ₐ - by sheldonison - 04/21/2015, 08:23 AM RE: How it looks (i.θ)ₐ - by marraco - 04/23/2015, 04:52 PM RE: How it looks (i.θ)ₐ - by JmsNxn - 04/23/2015, 11:15 PM RE: How it looks (i.θ)ₐ - by sheldonison - 04/23/2015, 11:20 PM RE: How it looks (i.θ)ₐ - by marraco - 04/26/2015, 12:50 AM RE: How it looks (i.θ)ₐ - by sheldonison - 04/26/2015, 05:08 AM RE: How it looks (i.θ)ₐ - by marraco - 04/20/2015, 03:46 AM RE: How it looks (i.θ)ₐ - by marraco - 04/23/2015, 02:21 AM RE: How it looks (i.θ)ₐ - by marraco - 04/25/2015, 07:52 PM

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