superfunctions of eta converge towards each other

[sheldonison]

Continuing with the conjecture, \( \text{sexp}_\eta(z) = \text{cheta(z+k+\theta(z)) \), here are some more graphs, and then some comments. cheta(z) is the upper entire superfunciton of eta.

\( \theta(z) \) from z=-1+i to z=1+i, which I previously posted, where theta(z) is well behaved, and decaying as imag(z) increases.
   
Next, lets regenerate this graph with \( \Im(z)=0.001 \), which is closer to the real axis, and closer to the singularity at ineger values of z.
   

To get closer to the real axis, we need to switch to a graph of the contour of \( \text{cheta(z+k) \), by graphing the contour of
\( \text{cheta}^{[-1]}(\text{sexp}_\eta(z))-\Re(k) \). The range if this graph is identical to the previous graph, from z=-1+0.001i to 1+0.001i. Only the real part of the constant k has been included, so the imag(z) values represent the contour values of cheta(z).
   

This last cheta(z) contour plot is a graph of the Kneser mapping contour for cheta(z), at the real axis itself, calculating the contour \( \text{cheta}^{[-1]}(z)-k \), with a range of z from \( z=-10^{11}+e\pi i \) to \( z=+10^{11}+e\pi i \). This is equivalent to the range of z from \( \text{sexp}_\eta(-2+\delta_a) \) to \( \text{sexp}_\eta(-3-\delta_b) \), where delta_a and delta_b are very small. Near -2, \( \delta_a \approx 10^{-1634000000} \). I think that the \( \delta_b \approx 10^{-\exp(1634000000)} \)... messy arithmetic. The graph has been modified so that it matches the range from -1 to +1 from above. For base e, I have observed that the graph of the Knser mapping contour continues becoming more and more complex, as we superexponentially approach the singularity. Similar complexity may occur for base eta. I may post more in the future.
   

Tommy wrote:

another question is : how many superfunctions can a function have ?

in this thread we have a lower and upper superfunction.

but when considering complex numbers and non-real fixpoints and general analytic functions , i wonder about how many superfunctions one can have and how to determine them.

Hey Tommy. There are some restrictions. These Kneser/Riemann mappings involve
1) a theta(z) function quickly decaying to zero at +imag infinity,
and
2) a resulting function with singularities at the integer values, where the singularities results in a Schwarz reflection, which allows the function to be defined for imag(z)<0.

We already have the regular superfunction for base e, as an example, which is Kneser mapped to produced sexp_e(z). This is another example, where the upper/entire superfunction, cheta(z) is Kneser mapped to produce sexp_eta(z).

These two restrictions, limit the kinds of functions involved. For tetration, this works for bases>eta, using the standard Kneser mapping, and for base cheta, as is conjectured.

For bases<eta, other theta(z) mappings are possible. I made an entire post about them last year, where I discussed base 2. http://math.eretrandre.org/tetrationforu...hp?tid=515
I have to refresh my memory on what I've posted, but I also derived a new different tetration solution for each base less than eta, using a Kneser mapping.
- Sheldon

[sheldonison]

.....
I originally thought I would calculate theta(z) by generating a Kneser Riemann mapping, which would require some modifications to my Kneser.gp code, that I haven't had time to try yet. So yesterday I calculated \( \theta(z) \), the easier way, using the equation.
\( \theta(z)=\text{cheta}^{[-1]}(\text{sexp}_\eta(z))-z \).....

This time, I calculated the Kneser Riemann mapping from cheta(z), which is the upper super-exponential of eta, calculating \( \text{sexp}_\eta(z) \), and theta(z) using my iterated Kneser/Riemann mapping algorithm \( \text{sexp}_\eta(z) = \text{cheta(z+\theta(z)) \)
Here are the first 80 terms of the resulting \( \text{sexp}_\eta \)taylor series, centered at 0, accurate to nearly 50 decimal digits, which is comparable to the accuracy of the cheta series from which it was generated. This required approximately 20 iterations of the iterative Kneser mapping algorithm.

Code:

a0=   1
a1=   0.611095453771651725262810441596468781580269047357550
a2=  -0.231702614476767880057754637266321761121523435531760
a3=   0.0917812876620532822133071038897611328881116958928124
a4=  -0.0375649217050071576220508834372714135192894502813784
a5=   0.0157737222017623151244855920986092095582090932573677
a6=  -0.00676146377741411160378256866430459401065680422398120
a7=   0.00294774761901529095651004530936485553141374655276524
a8=  -0.00130329397044496937265810128322368856058033652210951
a9=   0.000583068821879779779750519504424021334792203212824801
a10= -0.000263475538289731186662273991418485660413024460182638
a11=  0.000120079821309960737865224745056228269329158138384658
a12= -0.0000551303455271971740777327708851583454767974089869403
a13=  0.0000254727117285946134566265036623587110503043495453153
a14= -0.0000118350189839471092554362143395131603304332488176578
a15=  0.00000552555661721659732404978764658822685255716631441388
a16= -0.00000259088250250205448643621195471018577614859681535859
a17=  0.00000121947772793691463866748710287359617069950224343485
a18= -0.000000575938244261633170948573966605515258622439917077526
a19=  0.000000272835731915044808130761045812419754647293505221581
a20= -0.000000129604098503712840293865769356787634917932600346819
a21=  0.0000000617184648746551109652382067118582400221072683266451
a22= -0.0000000294572386417802551977422981166350346746812044161570
a23=  0.0000000140884643093386428902074176969660501764922998188352
a24= -0.00000000675079178733813416073761444717751772347711978606630
a25=  0.00000000324040197028192052818961738454119586307326826755909
a26= -0.00000000155789250617035227640871356618398318683270179424542
a27=  0.000000000750098596635507780777134988279191034195969015855036
a28= -0.000000000361655381928347049065842422084443198416688544391361
a29=  1.74592476875241231046123466193763330740029059024297 E-10
a30= -8.43864351753681771955715994960529383687218538708627 E-11
a31=  4.08321688631192290465015221458217806927196611211009 E-11
a32= -1.97780890956079928000397935766378593927042952963150 E-11
a33=  9.58937887210731107030786818689412786447899880426182 E-12
a34= -4.65366990984925474622297498839447451648951394617092 E-12
a35=  2.26035419763439608928450172822433746171836295149080 E-12
a36= -1.09878336817672683270058936312912776149186915209576 E-12
a37=  5.34543285201212110206484945017316458298407068619223 E-13
a38= -2.60238186381938710165111994894080921153824458691605 E-13
a39=  1.26782708786448072052855802829491283561033763740595 E-13
a40= -6.18065713730427984232303419660068094462142609552625 E-14
a41=  3.01495472825825957301602049941641113658174693814225 E-14
a42= -1.47158505471116819772042780418571488834513678652714 E-14
a43=  7.18681076071295928780918874220139590820603352332199 E-15
a44= -3.51173708545661500664450320814947737493335672976064 E-15
a45=  1.71684924476025713957768891113604892326056846887684 E-15
a46= -8.39763218512702291011716365584708191307710741717839 E-16
a47=  4.10947958309355559866994137817039774676100895009521 E-16
a48= -2.01193271357506415016943567634048365900544091276272 E-16
a49=  9.85436431469814660806778049713979975055654894405037 E-17
a50= -4.82863851527806912924306619834010899289430282129108 E-17
a51=  2.36697966470193428295397824205341791955432902543780 E-17
a52= -1.16073041261193125343825164385537017757777134584090 E-17
a53=  5.69414919431601466980627541017375839178407746579289 E-18
a54= -2.79435099362766606225911963240953587959485181378627 E-18
a55=  1.37177230600201211382050307050494974722622608593200 E-18
a56= -6.73638185995954869619382195705443352315783218342629 E-19
a57=  3.30909986107458123646909880258274197989614291968962 E-19
a58= -1.62602320761063609954141455195401222493393280029547 E-19
a59=  7.99231746118155302579947571730532923235669071272187 E-20
a60= -3.92955608509546808441160057820325985041095375019077 E-20
a61=  1.93256856644514224462909823307272388335074142227301 E-20
a62= -9.50699052849564802556568302196070479320577453210153 E-21
a63=  4.67804295847118247466699947260215068151288567726025 E-21
a64= -2.30247426862419264122607788485950688773998006169473 E-21
a65=  1.13352579378475997563046656633740111440713354501555 E-21
a66= -5.58175580272975796962068914975815305014506076078419 E-22
a67=  2.74922300731523716070763631069703995807246909834284 E-22
a68= -1.35439662860401999858849559830664314463541594037131 E-22
a69=  6.67383845978854646769324758362458777997658711938811 E-23
a70= -3.28924895518170155123068794530223425760681766093393 E-23
a71=  1.62146075255442663131012653517379778869865486264882 E-23
a72= -7.99470232162273878609029211157712241122041145401289 E-24
a73=  3.94259292573183323472088911673635062652752328594458 E-24
a74= -1.94465732147586014876037988016159230543477617405982 E-24
a75=  9.59364278594765374296940813010099907533150668148673 E-25
a76= -4.73370532201367019577101005937731419703884318213035 E-25
a77=  2.33611431476000136896220802065677131157325649743031 E-25
a78= -1.15308206561872134295524237335114322979399731394009 E-25
a79=  5.69243045052281048821530522001513494796180617476246 E-26
a80= -2.81063753494564060709822549726337867231556147101730 E-26

[tommy1729]

perhaps not so relevant but the following idea inspires me :

sexp(f(z)) = cheta(f(z))

f(z) satisfies f(z) = f(z) + theta(f(z)) + k

hence theta(f(z)) = -k

but theta is not a constant function , thus there is no f(z) apart from id(z).

[sheldonison]

... Here, sexp(z) is the lower superfunction, with sexp(0)=1, and cheta(z) is the upper superfunction ....
\( \text{sexp}_{\eta}(z)=\text{cheta}(z+\theta(z)+ k )
\). Where \( \theta(z) \) is a 1-cyclic function, which quickly decays to zero as imag(z) increases. Then, the constant "k" is 5.0552093131039000 + 1.0471975511965977*I ...

I made lots of minor updates and clarifications all over this reply.
Apparently, when I posted this last year nobody noticed that the imaginary part of k=1.0471975511965977 is exactly Pi/3, which of course begs for an explanation! I didn't notice it either, until I started to work with the formal Abel series solution for iterates of \( \exp(x)-1 \), which is parabolic with a fixed point of zero.

We start by noticing that solutions of \( g(z)=\text{cheta}(z)=\exp^{[z]}_\eta(2e) \) are conjugate to solutions for \( h(z)=(\exp(1)-1)^{[z]} \) so that \( h(z) = \frac{g(z)}{e}-1 \). So the two problems are trivially interchangeable. On mathstack, Will Jagy explained how to generate the formal abel function solution for the parabolic case. There are also papers by Baker on the abel function of exp(z)-1. Here is the formal solution for the abel function of exp(z)-1. I posted more terms below.
\( \alpha(\exp(z)-1)=\alpha(z)+1 \)
\( \alpha(z) = \frac{-2}{z} + \frac{\log(z)}{3} + \frac{-z}{36} + \frac{z^2}{540} + \frac{z^3}{7776} + \frac {-71z^4}{435456} + \frac{8759z^5}{163296000} + O(z^6) \)
\( \alpha(z)=\text{cheta^{-1}(e(z+1))+k \) for \( k\approx -2.025912 \)

The formal abel solution is only valid for real(x)>=0. It is also a divergent series, meaning that you need to truncate to some optimal number of terms for any particular value of z, but it is nonetheless very accurate. The 40 term series posted below was generated in pari-gp and is accurate to 32 decimal digits for |z|<=0.15. For larger values of z, iterate log(z+1) until the value is smaller than 0.15, and then evaluate the formal series. There is an analogous abel function for sexpeta.

\( \alpha_2(z)=\text{sexpeta^{-1}(e(z+1))+k \) for \( k\approx 3.029297 \)
\( \alpha_2(z) = \frac{-2}{z} + \frac{\log(-z)}{3} + \frac{-z}{36} + \frac{z^2}{540} + \frac{z^3}{7776} + \frac {-71z^4}{435456} + \frac{8759z^5}{163296000} + O(z^6) \)

This nearly identical \( \alpha_2(z) \) abel function is for the "sexpeta" superfunction of exp(z)-1. \( \alpha_2^{-1}(z) \) approaches zero from the negative real numbers, as z goes to infinity. \( \alpha_2(z) \) is only valid if real(z)<=0 . In this series, the log(z) term was replaced with log(-z), so that the abel function is real valued at the real axis for negative real numbers. For example, if we ignore the fact that \( \alpha(z) \) is not valid for \( \Re(z)<0 \), then \( \alpha(-0.01)\approx198.465+\frac{\pi i}{3} \), whereas \( \alpha_2(-0.01)\approx198.465 \). That \( \pi i/3=\log(-1)/3 \) difference between the two functions is exactly the imaginary part of the \( \theta(z) \) constant term for the two superfunctions of \( \exp_\eta(z) \), that I numerically calculated last year. If z=0.15i, than for the 40 term series posted below, both abel functions are valid, and should be accurate to >30 decimal digits. The two approximations differ by exactly \( \frac{\pi i}{3} \). But, as |z| grows, the formal solution is no longer very accurate, and one must iterate exp(z)-1 for \( \alpha_2(z) \), and iterate log(z+1) for \( \alpha_2(z) \), until each |z| is a smaller number before evaluating the formal solution. This iteration leads to the two inverse abel functions (superfunctions) behaving very differently as z approaches the real axis. But as imaginary of z increases, the inverse of the two functions converge towards each other.

\( \lim_{z \to \Im \infty}\hspace{2 mm}\alpha(\alpha_2^{-1}(z))-z = \pi i/3 \), which leads to the \( \theta(z) \) function I calculated. My definition for theta is \( \theta(z)=\alpha(\alpha_2^{-1}(z))-z \). The formal abel series solution may allow one to prove the exponential convergence as \( \Im(z) \) increases, which is conjectured to be: \( \alpha(\alpha^{-1}_2(z))-z=\pi i/3 + \sum_{n=1}^{\infty}a_n \exp(2n\pi zi) \). I've wanted to understand Ecalle cylinders for awhile ....

If anyone wants the pari-gp code for parabolic abel solutions for the general case, for \( f(x)=x+ax^2+... \), I could also post that. Also, I assume there is no equivalent formal solution for the superfunction, \( \alpha^{-1}(z) \), for the parabolic case. The best reasonable approximation I could generate for the reciprocal of the superfunction of exp(z)-1 is: fixed typo, updated approximation with emperical error bounds
\( \frac{1}{\alpha^{-1}(z)} \approx \frac{-z}{2}+\frac{-1}{6}\log(\frac{-z}{2})+O(z^{-1}) \)
\( \frac{1}{\alpha_2^{-1}(z)} \approx \frac{-z}{2}+\frac{-1}{6}\log(\frac{z}{2})+O(z^{-1}) \)
The \( O(z^{-1}) \) term seems to be \( (\frac{\log(\frac{-z}{2})}{18}-\frac{1}{36})\times z^{-1} \)
- Sheldon

Code:

First 30 terms, formal abel series term for exp(z)-1. log(z)/3 term also required
a-1= -2
a0=   0
a1=  -1/36
a2=   1/540
a3=   1/7776
a4=  -71/435456
a5=   8759/163296000
a6=   31/20995200
a7=  -183311/16460236800
a8=   23721961/6207860736000
a9=   293758693/117328567910400
a10= -1513018279/577754311680000
a11= -1642753608337/3355597042237440000
a12=  3353487022709/1689531377909760000
a13= -11579399106239/40790114695249920000
a14= -254879276942944519/137219514685385932800000
a15=  13687940105188979843/14114007224782553088000000
a16=  215276054202212944807/100956663443150497382400000
a17= -2657236754331703252459529/1203529624071657866919936000000
a18= -146435111462649069104449/50302321749125019205632000000
a19=  715411321613253460298674267/135588231530708185101474201600000
a20=  16634646784735044775309724063/3702250880735601413534515200000000
a21= -104353470644496360229598950087621/7332274212470670094037711585280000000
a22= -1026800310866887782669304706891/145015557324117535367532380160000000
a23=  10532451718209319314810847524219487/239106170881428081691713129676800000000
a24=  426818206492321153424287945331450731/55748747292256998858987528725200896000000
a25= -209820349077359397909291778326518401351/1340114117602331703341046363586560000000000
a26=  525117796674628883106100578152841570958289/21674067658217791337645745152194510848000000000
a27=  196370501349536911290241763355698126325788423/308676831703848984325152590299385561088000000000
a28= -4655964318554330930550687915598236845144401499/14371804056954685277851548955241890185216000000000
a29= -9047134015490968185454900363573980634933739699733371/3082105239034217031261653931196399559670497280000000000
a30=  205360181531874254884259531693649741510468924878159/77138340242791861182855632315816451715891200000000000
a31=  0.0152680842325604720475463799792485
a32= -0.0208547307456560124435878815421207
a33= -0.0888426680278022904549764943201458
a34=  0.169545350486845480899257484416871
a35=  0.573829524409951440465435202028753
a36= -1.47178737132655819000458397068965
a37= -4.08073498995482601179218114027270
a38=  13.7938917842810175925883583879899
a39=  31.6710432837078833141996786165051
a40= -140.151276797120726823402361527161
a41= -265.538550008913692150238411154415