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Chapter Nine
Taylor and Laurent Series
9.1. Taylor series.
Suppose
f
is analytic on the open disk 
z
z
0

r
.Let
z
be any point in
this disk and choose
C
to be the positively oriented circle of radius
,where

z
z
0

r
. Then for
s
C
we have
1
s
z
1
s
z
0
z
z
0
1
s
z
0
1
1
z
z
0
s
z
0
j
0
z
z
0
j
s
z
0
j
1
since 
z
z
0
s
z
0

1. The convergence is uniform, so we may integrate
C
f
s
s
z
ds
j
0
C
f
s
s
z
0
j
1
ds
z
z
0
j
,or
f
z
1
2
i
C
f
s
s
z
ds
j
0
1
2
i
C
f
s
s
z
0
j
1
ds
z
z
0
j
.
We have thus produced a power series having the given analytic function as a limit:
f
z
j
0
c
j
z
z
0
j
, 
z
z
0

r
,
where
c
j
1
2
i
C
f
s
s
z
0
j
1
ds
.
This is the celebrated
Taylor Series
for
f
at
z
z
0
.
We know we may differentiate the series to get
f
z
j
1
jc
j
z
z
0
j
1
9.1
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View Full Documentand this one converges uniformly where the series for
f
does. We can thus differentiate
again and again to obtain
f
n
z
j
n
j
j
1
j
2
j
n
1
c
j
z
z
0
j
n
.
Hence,
f
n
z
0
n
!
c
n
,o
r
c
n
f
n
z
0
n
!
.
But we also know that
c
n
1
2
i
C
f
s
s
z
0
n
1
ds
.
This gives us
f
n
z
0
n
!
2
i
C
f
s
s
z
0
n
1
ds
,fo
r
n
0,1,2,
.
This is the famous
Generalized Cauchy Integral Formula.
Recall that we previously
derived this formula for
n
0 and 1.
What does all this tell us about the radius of convergence of a power series? Suppose we
have
f
z
j
0
c
j
z
z
0
j
,
and the radius of convergence is
R
. Then we know, of course, that the limit function
f
is
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 Fall '09
 Peter

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