Taylor series of upx function

[Kouznetsov]

Dmitriy, can you please give a mathematica notebook for your method and base Sqrt(2)?

b=Sqrt[2]

RGB[x_,y_,z_]=RGBColor[x,y,z]

M=9

f=2-y+Sum[A[n]*y^n,{n,2,M}]

g=Series[ReplaceAll[f,y\[Rule]y*Exp[q]],{y,0,M}]

h=Simplify[Series[Exp[Log[b]*f],{y,0,M}]]

u=g-h



c1=Coefficient[u,y]

s1=Solve[c1\[Equal]0,q]

s11=Extract[s1,1]

Q=ReplaceAll[q,s11]

v[1]=Simplify[ReplaceAll[u,s11]]

For[i=2,i<M,
s[i]=Coefficient[v[i-1],y^i];
t[i]=Extract[Solve[s[i]\[Equal]0,A[i]],1];
Print[t[i]];
v[i]=ReplaceAll[v[i-1],t[i]];
i++]

a[0]=2; a[1]=-1;

For[i=2,i<M, a[i]=N[ReplaceAll[A[i],t[i]],18] ;i++]

For[i=0,i<M, Print[a[i]] ;i++]

F[z_]=Simplify[Sum[a[i]*Exp[Q*i*z],{i,0,M-1}]]

Plot[F[z],{z,-1,2}];

H[1,z_]=Log[F[z+1]]/Log[b]

For[i=2,i<14, H[i,z_]=Log[H[i-1,z+1]]/Log[b];i++]

Plot[{H[11,z]-H[12,z]},{z,-.5,2},PlotRange\[Rule]All];

z12=ReplaceAll[z,FindRoot[H[12,z]\[Equal]1,{z,-.5,1}]]

1.251551478822122`

J[z_]=H[12,z12+z]

P=Plot[{-z,J[z]},{z,-1.99,4},AspectRatio\[Rule]Automatic,
GridLines\[Rule]{{-2},{2}},PlotRange\[Rule]{{-2,4},{-4,2}}];

Export["tetrationQ2r.jpg",P]