Expanding Functions as Infinite Compositions
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Transcript of Expanding Functions as Infinite Compositions
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G 5 2013
( D)
:
:=
= 1 2
1
( ) ( )n
k nk
t z t t t z RRRR , =
=1
( ) lim ( )n
kn
k
T z t z RRRR .
:
=
= 1 1
1
( ) ( )n
k n nk
t z t t t z LLLL , =
=1
( ) lim ( )n
kn
k
T z t z LLLL .
C 1 2. H,
.
nz z . C :
= 22 ( )
(2 )1 ( )
Tan zTan z
Tan z . ( 2):
= = = =
=
2
2 2 42 2 2 21 14 4
42 21 14 4
( ) ( ) 2 ( ) 2 ( ) 4 ( )1 1 1 1
= 4 ( )1 1
z z z
z
z z z z T z Tan z Tan z Tan Tan
z z z z
z z z Tanz z
=
= = 1 2
1
( ) ( ) ( ( ))n
k n nk
T z t z t t t r z RRRR
= = 21
4
( ) , ( ) 2 ( / 2 )1 k
n n
k n
zt z r z Tan z z
z
C
= =
= =1 1
( ) lim ( ) ( )n
k kn k k
Tan z t z t z
R RR RR RR R
.
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1 1( ) ( ) , ( ) ( )k k n ng z t z z r z
= = (E ). ( ) , ( )k ng z z z z
1 1 1 1tan( ) ( ) ( )n n n n nArc z g g g z g g g z =
( )2142 4
( ) 1 1kk
kg z zz
= + .
1 1( ) ( )n n nG z g g g z = .
H, ( ) tan( )nG z Arc z ,
( )
=
= + 21
41
2 4tan( ) 1 1k
k
k
Arc z z zLLLL
, .. ,
. :
0
2tan( ) , 41 1
k kk k
z zArc z z
=
= = + + LLLL
.
C 0z=
1 n
{g } S=(|z| . , ( )0 0z S z R = < , n n n 1 1G (z) g g g (z) G(z)= ,
0S .
H
3
21
4
1( )
4 1 1 nn n
zg z z
z =
+ + . .
( ) 1zF z e= . (2 ) ( )( ( ) 2)F z F z F z = +
( )
2 2
2 2 22
( ) ( 2) ( / 2) 2 ( / 2) 2 ( / 2)4 4
= 2 4 ( / 4) 4 ( / 4)4 4 8
z zF z z z F z z z F z z F z
z z zz z z F z z z z F z
= + = + = +
+ + = + + =
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()
=
= +
2
1
1( )
10kk
zF z
zRRRR =5 8
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( ) ( )F z Sin z =
2 2 2 2
2 22
1 1( ) 2 1 ( / 2) 1 2 ( / 2) 1 4 1 ( / 4)
4 4
1 1 = 1 1 4 ( / 4)4 4
Sin z z z Sin z z z Sin z z z z z Sin z
z z z z Sin z
= = =
=
( ) 2 ( / 2 )n nnr z Sin z z = .
2
1
1( ) 1
4kkSin z z z
=
=
RRRR
1 4 2 3. F (1 4 )Sin i +
20n = .
() () =10
(CF) (), .
1
2
3
( )( )
( )1( )
11
a zF z
a za z
=
+
++
, ( )
( ; )1
n
n
a zt z
=
+
1 1( ; ) ( ; )T z t z = ,
1( ; ) ( ; ( ; ))n n nT z T z t z = .
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( ) lim ( ; )nn
F z T z
= ( )0
1
( ; )nn
t z
==
RRRR .
A 0 = , ( )F z .
CF
z ( )F z ,
, ( )F z .
( )
( ; )1
nn
a zt z
=
+ R <
1
2R
( )( ) 1na z R R< , 0 1< < ,
( )na z z S . ( ; )nt z R < .
1
( )( )
1n
n
a zF z
=
= +
RRRR , R < z S ( C 1).
C :
( ), ( ) 0f = ( ), 0 , ( )z z f = = .
4:
: , ,( ) ( )k n k ng z z z = + z S , ( )k ng z S S
. ,
lim 0k nn
= , () 1,2,...,k n= . 1, 1,( ) ( )n nG z g z=
, ( ), , 1,( ) ( )k n k n k nG z g G z= ,( ) ( )n n nG z G z= ( ) lim ( )nn
G z G z
= , .
. , , F=F ,
( ) ( )z F z z = , ( )G z = , F.
( )( ), ,( ) ,k n k ng z z F z z = + , 1F
z
