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 Abelfunction 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
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