08/28/2016, 06:20 PM
(This post was last modified: 08/29/2016, 11:55 AM by sheldonison.)

(08/28/2016, 03:13 PM)Xorter Wrote:(08/24/2016, 03:32 PM)sheldonison Wrote: ....

There is a formal asymptotic series for the Abel function solution for the parabolic case. Iterating , is congruent through a simple linear transformation to iterating cheta , by mapping

\alpha(z) = \frac{-1}{2z} + \frac{\ln(z)}{3} - \frac{z}{36} + \frac{z^2}{540} + \frac{z^3}{7776} + \frac{-71z^4}{435456} + ....

\alpha(\exp(z)-1) = \alpha(z)+1

[/tex]

The formal series will work in either half plane, by changing the ln(z) to ln(-z). But it will not work in both at the same time. It helps to iterate (exp(z)-1) or ln(z+1) a few times to get closer to the fixed point of zero before evaulating the asymptotic series. See G Edgar's post in mathoverflow for some theoretical background on the parabolic case. http://mathoverflow.net/questions/45608/...x-converge

Hello, Sheldon!

I tried your series for abel function. But it does not seem it would be the inverse of cheta function. Why not? Did I make a mistake or what?

(Here is the picture of the graphs.)

I only posted four terms of the series. Also, the series is an asymptotic series so the optimal number of terms varies. But if you iterate enough times until z is fairly closed to the fixed point of zero; say within 0.1 with a 25 term series, then the series gives excellent results. Below is a graph of 25 terms series, with zk=2.02591209868586388250227776560583118127388887; plotting from 0.02 to 0.1. Notice the Kneser.gp invcheta gives nearly identical results within 10^-32 of the formal parabolic series, even though the kneser.gp code uses a completely different algorithm since I didn't even know about the formal parabolic series when I wrote the Kneser.gp program. The equation for the function I graphed is:

where

Code:

`zk=2.02591209868586388250227776560583118127388887;`

newabel(z) = { (1/3)*log(z) + subst(parabolic,x,z); }

ploth(t=0.02,0.1,z=cheta(newabel(t)+zk)/exp(1)-1-t)

{parabolic=

(1/x)* -2

+x^ 1* -1/36

+x^ 2* 1/540

+x^ 3* 1/7776

+x^ 4* -71/435456

+x^ 5* 8759/163296000

+x^ 6* 31/20995200

+x^ 7* -183311/16460236800

+x^ 8* 23721961/6207860736000

+x^ 9* 293758693/117328567910400

+x^10* -1513018279/577754311680000

+x^11* -1642753608337/3355597042237440000

+x^12* 3353487022709/1689531377909760000

+x^13* -11579399106239/40790114695249920000

+x^14* -254879276942944519/137219514685385932800000

+x^15* 13687940105188979843/14114007224782553088000000

+x^16* 215276054202212944807/100956663443150497382400000

+x^17* -2657236754331703252459529/1203529624071657866919936000000

+x^18* -146435111462649069104449/50302321749125019205632000000

+x^19* 715411321613253460298674267/135588231530708185101474201600000

+x^20* 16634646784735044775309724063/3702250880735601413534515200000000

+x^21* -104353470644496360229598950087621/7332274212470670094037711585280000000

+x^22* -1026800310866887782669304706891/145015557324117535367532380160000000

+x^23* 10532451718209319314810847524219487/239106170881428081691713129676800000000

+x^24* 426818206492321153424287945331450731/55748747292256998858987528725200896000000

+x^25* -209820349077359397909291778326518401351/1340114117602331703341046363586560000000000

}

- Sheldon