Hyperbolic Fractional Iteration
Hyperbolic Fractional Iteration
H[0]=0;H[1]=f'[0]^t;Do[H[max]=First[r[t]/.RSolve[{r[0]==0,r[t]==Sum[Derivative[k][f][0]BellY[max,k,Table[H[j]/.t->t-1,{j,max}]],{k,2,max}]+f'[0]r[t-1]},r[t],t]],{max,2,6}]
Validation
Validation
Show ((z))-(z)=0
a
f
b
f
a+b
f
In[]:=
s=f'[0]^tz+Sum[1/k!H[k]z^k,{k,2,6}];sa=s/.t->a;sb=s/.t->b;sc=s/.t->a+b;Series[(sa/.z->sb)-sc,{z,0,6}]
Out[]=
7
O[z]
Parabolic Fractional Iteration
Parabolic Fractional Iteration
In[]:=
p=Normal[Series[Limit[s,{f'[0]->1}],{z,0,6}]];
In[]:=
pa=p/.t->a;pb=p/.t->b;pc=p/.t->a+b;Series[(pa/.z->pb)-pc,{z,0,6}]
Out[]=
7
O[z]