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Introducing myself ...
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:
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.... Sad
Maybe I'm totally wrong with some key properties of this formula?

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

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 Sad

Prog is inside ZIP with some other of my tetration attempts is linear and quadric approximation is Newton approx is opengl renderer of results

There are binaries too for FreeBSD UNIX 10-CURRENT
I any of You want then You can remotely login to my home server through SSH
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
Br0 --> Br1, step Bri (iterate base in real direction)
Bi0 --> Bi1, step Bii (iterate base in imag direction)
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
Lots of lines

New program handles this, and outputs to such a file.
Values are strange and I don't know how to fix it. will generate such values for exponent b^x, this is for testing render4. New
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.

Attached Files
.zip (Size: 70.77 KB / Downloads: 462)
.zip (Size: 8.35 KB / Downloads: 471)
Fuji GSW690III
Nikon D3, Nikkors 14-24/2.8, 24/1.4, 35/2, 50/1.4, 85/1.4, 135/2, 80-200/2.8
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 is not defined, and instead the Schroder function has a singularity at zero.

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>, the fixed point is complex, so a Riemann mapping is required to modify the regular superfunction, 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 , where is a 1-cyclic function which decays to 0 as increases and is equivalent to the Riemann mapping. Of course, 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
+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

BTW: after reading this:
All is clear until post #18 - I'm exactly on that point:
mike3: "Now all we need is some way to express the Riemann mapping"
I have no proble with bell polynomials, determinant, matrices, fixed points for any base, periods, etc - all steps which made final expression for a_n (post #17) is clear for me, but I thought initially that this is the end - that this will give me final method for approximation, but now seems that I'm on half-way at max.

Updated 1st post, added few new progs - but no real progress in tetration so far.
Fuji GSW690III
Nikon D3, Nikkors 14-24/2.8, 24/1.4, 35/2, 50/1.4, 85/1.4, 135/2, 80-200/2.8

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