# Tetration Forum

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Hi everyone. My name is Lucas Gryglicki.
I was born in 16th July 1982.
I will have some questions for You, I'm browsing this forum since some years, and I'm a big fan of math and specially tetration is my main interest. I've written many program with successfull linear, quadric approx, but currently got stuck in analytical formula for any base.
I will ask for it later.

Hello to all of You.

My problem is:
From this forum: analytical formula for tetration:
http://math.eretrandre.org/tetrationforu...hp?tid=576
sum(0...inf) A_n * L^(Nz)

This seems to be for base E only, but I found formula for A_n for any base
I have algorithm to find L for b
I have algorithm to compute A_n -- bell complete of recursive previous A_n-1 with log(b) terms etc - it works and gives correct results, but when I finally try it even for base E and for z = -1,0,1,2 I should get 0,1,E,E^E but I get strange things .... Henryk told me that this formula has singularities in integer values, but this is new info for me, I was trying many many days to understand why it fails.
So is there *any* formula to compute sexp(z) for any base E - some infinite sum
I tried A_n * (z-L)^N, etc - nothing helps....
Maybe I'm totally wrong with some key properties of this formula?

If Anyone of You wants then please take a look at attached anal.cc
It can be easily compiled with gcc
g++ -o anal anal.cc

Many other of my progs uses MPFR, MPC, GMP itd, OpenGL - but this one is the simplest one it contains: LUP decomposition, determinant of NxN matrix, factorials, binonmials compute, bell complete polynomial, recurence formula for A_n.... and finite sum of A_n * ?? --> to get some simplest tetrations for lets say b=2 and for z = 0; 0.5; 1 - but it fails and I don't know why

Prog is anal.cc inside ZIP with some other of my tetration attempts
mpctet.cc is linear and quadric approximation
newton.cc is Newton approx
render.cc is opengl renderer of results

I any of You want then You can remotely login to my home server through SSH
IP: 88.199.169.19
Port: 9922

Login and Pass with private msgs only
It is quite a powerful server, with ZFS mirror pools, 16 GB of RAM, 4 CPUs....
can do many many math computation quite fast.

I've made some programs to see how current results look like.
Written generic program to display
Base:
Br0 --> Br1, step Bri (iterate base in real direction)
Bi0 --> Bi1, step Bii (iterate base in imag direction)
Height:
Xr0 --> Xr1, step Xri (iterate tetration height in real direction)
Xr0 --> Xi1, step Xii (iterate tetration in imag direction)

Finally we have 4 ranges as input and output is complex value (Cr, Ci)
CSV is c = b^^x
br_idx;bi_idx;xr_idx;xi_idx;br;bi;xr;xi;cr;ci;
Lots of lines

New program render4.cc handles this, and anal.cc outputs to such a file.
Values are strange and I don't know how to fix it.
genexp.cc will generate such values for exponent b^x, this is for testing render4. New zip:tet.zip
Also anal handles perdiodic of b^^x to fins point which is nearest to L, and then iterate +/- 1,2,3.... in real direction to get as close as possible to L, and using b^^x = b^(b^^(x-1)) or b^^x = log(b^^(x+1))/log(b) during each step.
Welcome Lucas!

You might try posting in the math and general discussion subsection with your question too, often I just look at the Tetration and related topics section. There are formulas for the schroder coefficients for base e, which has a complex fixed point. But since the Schroder function has a complex fixed point, L~=0.318 + 1.337i, then the function is not real valued at the real axis. The Schroder function has a singularity at zero, so the regular superfunction generated from the inverse Schroder function cannot give you the sequence that you are looking for. This sequence won't work since $\psi(0)$ is not defined, and instead the Schroder function has a singularity at zero.
$\psi^{-1}(\psi(0))=0$
$\psi^{-1}(L\psi(0))=1$
$\psi^{-1}(L^2\psi(0))=e$
$\psi^{-1}(L^3\psi(0))=e^e$

There is a closed form for the Schroder equation for the complex coefficients for base e, in the post you reference. There is another post here on the forum; see Mike's subpost #9.

But for real valued base e tetration, or for any base>$e^{1/e}$, the fixed point is complex, so a Riemann mapping is required to modify the regular superfunction, $\psi^{-1}(L^z)$ so that it becomes real valued. The Riemann mapping is complicated, and there is no simple closed form formula for real valued tetration, but there are accurate Taylor series approximations. The algorithm I use is $\psi^{-1}(L^{z+\theta(z)})$, where $\theta(z)$ is a 1-cyclic function which decays to 0 as $\Im(z)$ increases and is equivalent to the Riemann mapping. Of course, $\theta(z)$ has a nasty singularity at integer values. There is pari-gp code to generate the Taylor series for Kneser's algorithm here.

Again, welcome, and hope this answer helps a little bit. There is a Taylor series for tetration base e below.
- Sheldon
Code:
{tet(x)=         1.0000000000000000000 +x^ 1*  1.0917673512583209918 +x^ 2*  0.27148321290169459533 +x^ 3*  0.21245324817625628431 +x^ 4*  0.069540376139987373729 +x^ 5*  0.044291952090473304406 +x^ 6*  0.014736742096389391152 +x^ 7*  0.0086687818172252603664 +x^ 8*  0.0027964793983854596948 +x^ 9*  0.0016106312905842720722 +x^10*  0.00048992723148437733470 +x^11*  0.00028818107115404581135 +x^12*  0.000080094612538543333444 +x^13*  0.000050291141793805403695 +x^14*  0.000012183790344900091616 +x^15*  0.0000086655336673815746852 +x^16*  0.0000016877823193175389918 +x^17*  0.0000014932532485734925811 +x^18*  0.00000019876076420492745532 +x^19*  0.00000026086735600432637316 +x^20*  0.000000014709954142541901861 +x^21*  0.000000046834497327413506255 +x^22* -0.0000000015492416655467695218 +x^23*  0.0000000087415107813509359130 +x^24* -0.0000000011257873101030623176 +x^25*  0.0000000017079592672707284126 +x^26* -0.00000000037785831549229851765 +x^27*  0.00000000034957787651102163179 +x^28* -1.0537701234450015066 E-10 +x^29*  7.4590971476075052807 E-11 +x^30* -2.7175982065777348693 E-11 +x^31*  1.6460766106614471304 E-11 +x^32* -6.7418731524050529991 E-12 +x^33*  3.7253287233194685443 E-12 +x^34* -1.6390873267935902235 E-12 +x^35*  8.5836383113585680605 E-13 +x^36* -3.9437387391053843136 E-13 +x^37*  2.0025231280218870559 E-13 +x^38* -9.4419622429240650237 E-14 +x^39*  4.7120547458493713408 E-14 +x^40* -2.2562918820355970800 E-14 +x^41*  1.1154688506165369963 E-14 +x^42* -5.3907455570163504919 E-15 +x^43*  2.6521584915166818728 E-15 +x^44* -1.2889107655445536819 E-15 +x^45*  6.3266785019566604528 E-16 +x^46* -3.0854571504923359890 E-16 +x^47*  1.5131767717827405271 E-16 +x^48* -7.3965341370947514333 E-17 +x^49*  3.6269876710541876035 E-17 +x^50* -1.7757255986762984030 E-17 +x^51*  8.7098795443960546503 E-18 +x^52* -4.2692892823391563142 E-18 +x^53*  2.0950441625755281093 E-18 +x^54* -1.0278837092822587892 E-18 +x^55*  5.0468242474381763890 E-19 +x^56* -2.4780505958215521454 E-19 +x^57*  1.2173942030393317020 E-19 +x^58* -5.9816486323037815151 E-20 +x^59*  2.9402643445138969081 E-20 +x^60* -1.4455835436201850220 E-20 }
Hmmm, thanks - this of course helps - because I know now that my attempt was dead-end.
I'll try again to understand how this Riemann mapping works... because for now I don't know how it works, of course I can use ready formula - but this is not my way. I don't like using something that I don't understand.

Of course if You can point me to any source from where I can start trying to understand Riemann mapping - I've read wikipedia of course, but maybe I'm too "weka" to understand how to apply it to tetration.

Aditional question - are these A_n coefficients that I already computed useful for anything, or should I start from really beginning? Maybe I just need some kind of "transformation" of some terms in my formula?

Hmm I don't really know even how to ask for this what I want.
Ideal would be some kind of just explaination how to make Riemann mapping for let's say beginner - of course if anybody wants to help me with it.

Big thanks and happy christmas and new year for all