Taylor series of cheta

[sheldonison]

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?

Well its an asymptotic series; the theory is somewhat complicated and that would require dozens of pages to explain. For large values of |z|>1 the asymptotic doesn't converge at all and instead diverges! If you stick to values with |z|<0.1 with 25 terms, you'll get accuracy to 32 decimal digits. So lets say you want \( \alpha(100) \); then you need to iterate mapping \( z \mapsto \log(z+1) \) 20 times before evaluating the Abel function to get 32 decimal digits accuracy (iterating 10 times will give you 21 decimal digits accuracy). You get z~=0.109413515605671; and then increment \( \alpha(z)+20 \) to get the Abel function you want. Of course, before you even start you need to map \( z \mapsto \frac{z}{e}-1 \). Then here is the pari-gp code ...

Code:

invchetaabel(z) = {
local(n);
zk=2.02591209868586388250227776560583118127388887;
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;

z=z/exp(1)-1;
for (n=1,20,z=log(z+1));
return((1/3)*log(z) + subst(parabolic,x,z) + zk + 20);
}

The theory gets even more complicated for \( \alpha(z) \) for negative values of z, which would require another dozen or so pages to explain. But besides the upper superfunction analogous to cheta, there is another superfunction for iterating \( z \mapsto \exp(z)-1 \). And the two functions are completely different even though they have the same asymptotic Abel series with log(z) replaced with log(-z)! For that other superfunction, for negative values of z, you iterate \( z \mapsto \exp(z)-1 \) instead!