closed form for regular superfunction expressed as a periodic function - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: closed form for regular superfunction expressed as a periodic function (/showthread.php?tid=502) Pages: 1 2 3 4 closed form for regular superfunction expressed as a periodic function - sheldonison - 08/27/2010 I edited the title of this post to "closed form for regular superfunction..." Normally, we would look at the regular superfunction (base e) as a limit, $\lim_{n \to \infty} \exp^{[n]} (L+L^{z-n})$ but it has a period of 2Pi*i/L, so perhaps it could also be expressed as an infinite sum of periodic terms, $\sum_{n=0}^{\infty}a_n\times L^{( n*z)}$ $a_0=L$ and I think $a_1=1$. Perhaps there is a closed form limit equation for the other a_n terms in the periodic series? - Sheldon RE: regular superfunction expressed as a periodic function - sheldonison - 08/28/2010 (08/27/2010, 02:09 PM)sheldonison Wrote: .... Perhaps there is a closed form limit equation for the other a_n terms in the periodic series? - SheldonOriginally, I intended to solve it mathematically, with limit equations of some sort, but I gave up. Anyway, I brute forced the numerical solutions using complex Fourier analysis, and the results looked good. $a_0=L$ $a_1=1$ $a_2=1/(2*L-2)$ I suspect a_3 would be the reciprical of a polynomial in L, L^2, but I wasn't able to find the pattern... yet. I will continue to pursue both limit equations, and numerical results .... Here are the numerical results. This would provide another way of calculating the regular superfunction. 0.318131505204764 + 1.33723570143069*I 1.00000000000000 - 2.60599763775608 E-46*I -0.151314897155652 - 0.296748836732241*I -0.0369763094090676 + 0.0987305443114970*I 0.0258115979731401 - 0.0173869621265308*I -0.00794441960244236 + 0.000579250181689956*I 0.00197153171916544 + 0.000838273147502224*I -0.000392010935257457 - 0.000393133164925080*I 0.0000581917506305269 + 0.000119532747356117*I -0.00000315362731515909 - 0.0000302507270044311*I -0.00000144282204032780 + 0.00000712739202459367*I 0.000000659214290634412 - 0.00000152248373494640*I -0.000000194922185012021 + 0.000000284379660774925*I 0.0000000534780813335645 - 0.0000000461525820619042*I -0.0000000140401213835816 + 0.00000000762603302713942*I 0.00000000315342989238929 - 0.00000000146720737059747*I -0.000000000591418449135739 + 0.000000000254860555536866*I 1.07938974205158 E-10 - 2.18783515598918 E-11*I -2.47877770137887 E-11 - 1.92092983484722 E-12*I 6.07224941506231 E-12 + 1.28233399551471 E-13*I -1.11638719710692 E-12 + 2.71214581539618 E-13*I 1.27498904804777 E-13 - 5.65381557065319 E-14*I -1.63566889526104 E-14 - 9.67900985098162 E-15*I 7.43073077006837 E-15 + 4.93269953498429 E-15*I -2.43590387189014 E-15 - 5.21055989895944 E-17*I 3.19983418022330 E-16 - 3.06533506041410 E-16*I 2.35111634696870 E-17 + 4.77456396407459 E-17*I -7.31502044718856 E-18 + 1.10974529579819 E-17*I -3.26221285971496 E-18 - 3.99734388033735 E-18*I 1.39071827030212 E-18 - 2.43545980631280 E-19*I -8.14762787710817 E-20 + 3.02121739652423 E-19*I -5.68424637170392 E-20 - 3.30544545268232 E-20*I 1.00134634420191 E-20 - 1.20846493207443 E-20*I 2.44040335072592 E-21 + 3.32815768651263 E-21*I -1.00281770980843 E-21 + 3.16126717352352 E-22*I 7.46727364275524 E-24 - 2.52461237780373 E-22*I 5.59031612947253 E-23 + 1.92941756355972 E-23*I -8.03434017691000 E-24 + 1.15003875138068 E-23*I -2.13560370410329 E-24 - 2.64275348521218 E-24*I 7.66352536099374 E-25 - 3.06816449234638 E-25*I 1.42509561746275 E-26 + 1.97884314138339 E-25*I -4.58748818093499 E-26 - 1.10298477574201 E-26*I 5.73983689586017 E-27 - 9.59735542776205 E-27*I 1.77067703342646 E-27 + 1.97743801995778 E-27*I -5.74269137934548 E-28 + 2.60660071774678 E-28*I -1.76277133832552 E-29 - 1.48518580873011 E-28*I 3.46938198356513 E-29 + 6.74672128662828 E-30*I -4.01277862515764 E-30 + 7.29505202539481 E-30*I -1.34399669008111 E-30 - 1.42300404827861 E-30*I 4.14782554384895 E-31 - 1.97836607707266 E-31*I RE: regular superfunction expressed as a periodic function - tommy1729 - 08/28/2010 that seems efficient and intresting. in fact i doubt it hasnt been considered before ? RE: regular superfunction expressed as a periodic function - sheldonison - 08/30/2010 (08/28/2010, 11:21 PM)tommy1729 Wrote: that seems efficient and intresting. in fact i doubt it hasnt been considered before ?Thanks Tommy! I assume it has been considered, and probably calculated before. I think Kneser developed the complex periodic regular tetration for base e, and probably would've generated the coefficients. But I haven't seen them before. Perhaps Henryk (or someone else) could comment??? I figured out the closed form equation for a couple more terms, and I have an equation that should generate the other terms, but I'm still working it, literally as I write this post! $a_2 = (1/2)/(L - 1)$ $a_3 = (1/6 + a_2)/(L*L - 1)$ $a_4 = (1/24 + (1/2)*a_2*a_2 + (1/2)*a_2 + a_3)/(L*L*L-1)$ What I did is start with the equation: $\text{RegularSuperf}(z) = \sum_{n=0}^{\infty}a_nL^{nz}$ and set it equal to the equation $\text{RegularSuperf}(z) = \exp{(\text{RegularSuperf}(z-1))}$ Continuing, there is a bit of trickery in this step to keep the equations in terms of $L^{nz}$, instead of in terms of $L^{n(z-1)}$. Notice that $L^{n(z-1)}=L^{(nz-n)}=L^{-n}L^{nz}$. $\text{RegularSuperf}(z) = \exp{(\text{RegularSuperf}(z-1))} = \exp{( \sum_{n=0}^{\infty}\exp^{(L^{-n}a_nL^{nz})})}$ This becomes a product, with $a_0=L$ and $a_1=1$ $\text{RegularSuperf}(z) = \prod_{n=0}^{\infty} \exp{(L^{-n}a_nL^{nz})}$ The goal is to get an equation in terms of $L^{nz}$ on both sides of the equation. Then I had a breakthrough, while I was typing this post!!!! The breakthrough is to set $y=L^z$, and rewrite all of the equations in terms of y! This wraps the 2Pi*I/L cyclic Fourier series around the unit circle, as an analytic function in terms of y, which greatly simplifies the equations, and also helps to justify the equations. $\text{RegularSuperf}(z) = \sum_{n=0}^{\infty}a_ny^n = \prod_{n=0}^{\infty} \exp{(L^{-n}a_ny^n)}$ The next step is to expand the individual Tayler series for the $\exp {(L^{-n}a_ny^n)}$, and multiply them all together (which gets a little messy, but remember a0=L and a1=1), and finally equate the terms in $y^n$ on the left hand side equation with those on the right hand side equation, and solve for the individual $a_n$ coefficients. Anyway, the equations match the numerical results. I'll fill in the Tayler series substitution next time; this post is already much more detailed then I thought it was going to be! I figured a lot of this out as I typed this post! - Sheldon RE: regular superfunction expressed as a periodic function - Gottfried - 08/30/2010 (08/27/2010, 02:09 PM)sheldonison Wrote: Normally, we would look at the regular superfunction (base e) as a limit, $\lim_{n \to \infty} \exp^{[n]} (L+L^{z-n})$ but it has a period of 2Pi*i/L, so perhaps it could also be expressed as an infinite sum of periodic terms, $\sum_{n=0}^{\infty}a_n\times L^{( n*z)}$ $a_0=L$ and I think $a_1=1$. Perhaps there is a closed form limit equation for the other a_n terms in the periodic series? - Sheldon Hi Sheldon - just to allow me to follow (think I can't involve much) - I don't have a clue from where this is coming, what, for instance, is L at all? I think you've explained it elsewhere before but don't see it at the moment... Would you mind to reexplain in short or to provide the link? Gottfried RE: regular superfunction expressed as a periodic function - tommy1729 - 08/30/2010 (08/30/2010, 09:22 AM)Gottfried Wrote: (08/27/2010, 02:09 PM)sheldonison Wrote: Normally, we would look at the regular superfunction (base e) as a limit, $\lim_{n \to \infty} \exp^{[n]} (L+L^{z-n})$ but it has a period of 2Pi*i/L, so perhaps it could also be expressed as an infinite sum of periodic terms, $\sum_{n=0}^{\infty}a_n\times L^{( n*z)}$ $a_0=L$ and I think $a_1=1$. Perhaps there is a closed form limit equation for the other a_n terms in the periodic series? - Sheldon Hi Sheldon - just to allow me to follow (think I can't involve much) - I don't have a clue from where this is coming, what, for instance, is L at all? I think you've explained it elsewhere before but don't see it at the moment... Would you mind to reexplain in short or to provide the link? Gottfried L is the fixpoint of exp. we have been using L for 2 years RE: regular superfunction expressed as a periodic function - tommy1729 - 08/30/2010 i must say though that i dont completely get how " fourier methods" lead to the closed forms of a_n ... any fourier expert/sheldon want to explain ? further i believe regular superfunction means expanded at upper fixpoint or at least an equivalent of that. RE: regular superfunction expressed as a periodic function - sheldonison - 08/31/2010 (08/30/2010, 09:46 AM)tommy1729 Wrote: i must say though that i dont completely get how " fourier methods" lead to the closed forms of a_n ... any fourier expert/sheldon want to explain ? further i believe regular superfunction means expanded at upper fixpoint or at least an equivalent of that.I agree with Tommy when he says "Fourier methods" (complex Fourier analysis of a discreet number of points), won't lead to a closed form, and I didn't use Fourier methods to generate the coefficients for a_4, and a_5. Also, so far, I've only used this method for the regular superfunction for base(e), although it would presumably work for any regular periodic superfunction (using the upper fixed point, for bases