12/05/2012, 12:22 AM
(This post was last modified: 12/05/2012, 06:44 PM by sheldonison.)

(05/23/2011, 09:01 PM)sheldonison Wrote: ... Here, sexp(z) is the lower superfunction, with sexp(0)=1, and cheta(z) is the upper superfunction ....I made lots of minor updates and clarifications all over this reply.

. Where is a 1-cyclic function, which quickly decays to zero as imag(z) increases. Then, the constant "k" is 5.0552093131039000 + 1.0471975511965977*I ...

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 , which is parabolic with a fixed point of zero.

We start by noticing that solutions of are conjugate to solutions for so that . 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.

for

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.

for

This nearly identical abel function is for the "sexpeta" superfunction of exp(z)-1. approaches zero from the negative real numbers, as z goes to infinity. 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 is not valid for , then , whereas . That difference between the two functions is exactly the imaginary part of the constant term for the two superfunctions of , 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 . But, as |z| grows, the formal solution is no longer very accurate, and one must iterate exp(z)-1 for , and iterate log(z+1) for , 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.

, which leads to the function I calculated. My definition for theta is . The formal abel series solution may allow one to prove the exponential convergence as increases, which is conjectured to be: . 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 , I could also post that. Also, I assume there is no equivalent formal solution for the superfunction, , 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

The term seems to be

- 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