(08/28/2016, 06:20 PM)sheldonison Wrote: 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 \( z \mapsto \ln(z+1) \) 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:
\( \left(\frac{\text{cheta}(\text{AbelSeries}(z)+zk)}{e}\right)-1-z\;\;
zk=\text{cheta}^{-1}((0.02+1)\cdot e )-\text{AbelSeries}(0.02)
\)
Code:zk=2.02591209868586388250227776560583118127388887;
newabel(z) = { (1/3)*log(z) + subst(parabolic,x,z); }
ploth(t=0.02,0.1,z=\text{cheta}(\text{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
}
The more term I use in the graph, the less it similars to the inverse of cheta function (abel is the blue function). I used all your terms, but it does not work. Why cannot I get the inverse of cheta function from it?
Xorter Unizo
