Work

In[]:=
Quit[];
In[]:=
Clear["FractionalIteration`*"]​​Internal`$ContextMarks=False;​​<<FractionalIteration`​​max=4;
In[]:=
FractionalIteration[f,n,z,0,max]
Out[]=
z
n
′
f
[0]
+
1
2
2
z
-1+n
∑
k
1
=0
-2+2n-
k
1
′
f
[0]
′′
f
[0]+
1
6
3
z
3
-1+n
∑
k
1
=0
-2+n-
k
1
∑
k
2
=0
-5+3n-
k
2
-2
k
1
′
f
[0]
2
′′
f
[0]
+
-1+n
∑
k
1
=0
-3+3n-2
k
1
′
f
[0]
(3)
f
[0]+
1
24
4
z
3
-1+n
∑
k
1
=0
-2+n-
k
1
∑
k
2
=0
-2+n-
k
1
∑
k
3
=0
-8+4n-
k
2
-
k
3
-3
k
1
′
f
[0]
3
′′
f
[0]
+12
-1+n
∑
k
1
=0
-2+n-
k
1
∑
k
2
=0
-3+n-
k
2
-
k
1
∑
k
3
=0
-9+4n-
k
3
-2
k
2
-3
k
1
′
f
[0]
3
′′
f
[0]
+6
-1+n
∑
k
1
=0
-2+n-
k
1
∑
k
2
=0
-6+4n-
k
2
-3
k
1
′
f
[0]
′′
f
[0]
(3)
f
[0]+4
-1+n
∑
k
1
=0
-2+n-
k
1
∑
k
2
=0
-7+4n-2
k
2
-3
k
1
′
f
[0]
′′
f
[0]
(3)
f
[0]+
-1+n
∑
k
1
=0
-4+4n-3
k
1
′
f
[0]
(4)
f
[0]
In[]:=
HyperbolicIteration[f,n,z,0,max,Verbose->True]
Out[]=
z
n
′
f
[0]
+
2
z
-1+n
′
f
[0]
(-1+
n
′
f
[0]
)
′′
f
[0]
2(-1+
′
f
[0])
+
1
6
3
z
3
-2+n
′
f
[0]
(-1+
n
′
f
[0]
)(-
′
f
[0]+
n
′
f
[0]
)
2
′′
f
[0]
2
(-1+
′
f
[0])
(1+
′
f
[0])
+
-1+n
′
f
[0]
(-1+
2n
′
f
[0]
)
(3)
f
[0]
-1+
2
′
f
[0]
+
1
24
4
z
12
-3+n
′
f
[0]
(-1+
n
′
f
[0]
)(
3
′
f
[0]
+
2n
′
f
[0]
-
1+n
′
f
[0]
-
2+n
′
f
[0]
)
3
′′
f
[0]
3
(-1+
′
f
[0])
(1+
′
f
[0])(1+
′
f
[0]+
2
′
f
[0]
)
+(3
-3+n
′
f
[0]
(-1+
n
′
f
[0]
)(
′
f
[0]+
3
′
f
[0]
-
n
′
f
[0]
+
2n
′
f
[0]
-
1+n
′
f
[0]
-2
2+n
′
f
[0]
+
1+2n
′
f
[0]
)
3
′′
f
[0]
)(
3
(-1+
′
f
[0])
(1+
′
f
[0])(1+
′
f
[0]+
2
′
f
[0]
))+
4
-2+n
′
f
[0]
(-1+
n
′
f
[0]
)(-
′
f
[0]-
2
′
f
[0]
+
n
′
f
[0]
+
2n
′
f
[0]
)
′′
f
[0]
(3)
f
[0]
2
(-1+
′
f
[0])
(1+
′
f
[0])(1+
′
f
[0]+
2
′
f
[0]
)
+
6
-2+n
′
f
[0]
(-1+
n
′
f
[0]
)(-
2
′
f
[0]
+
2n
′
f
[0]
-
2+n
′
f
[0]
+
1+2n
′
f
[0]
)
′′
f
[0]
(3)
f
[0]
2
(-1+
′
f
[0])
(1+
′
f
[0])(1+
′
f
[0]+
2
′
f
[0]
)
+
-1+n
′
f
[0]
(-1+
3n
′
f
[0]
)
(4)
f
[0]
-1+
3
′
f
[0]
In[]:=
ParabolicIteration[f,n,z,0,max,Verbose->True]
Out[]=
z+
1
2
n
2
z
′′
f
[0]+
1
6
3
z
3
2
(-1+n)n
2
′′
f
[0]
+n
(3)
f
[0]+
1
24
4
z
2n(2-3n+
2
n
)
3
′′
f
[0]
+
1
2
n(1-3n+2
2
n
)
3
′′
f
[0]
+5(-1+n)n
′′
f
[0]
(3)
f
[0]+n
(4)
f
[0]
In[]:=
t=FractionalIteration[f,n,z,0,max];
In[]:=
ta=t/.n->a//ReleaseHold;​​tb=t/.n->b//ReleaseHold;​​tc=t/.n->(a+b)//ReleaseHold;​​Series[(ta/.z->tb)-tc,{z,0,max}]//Simplify
Out[]=
5
O[z]