Page 89 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes To prove this, begin by letting  > 0 and choosing N > J so that

n
 M  
j
j m

for all n, m > N. (We can do this because of the famous Cauchy criterion.) Next, observe that

n n n
 f (z)   f (z)   M   .
j
j
j
j m j m j m



 n 
This shows that  f (z) converges. To see the uniform convergence, observe that
 j 
 j 1 

n n m 1

 f (z)   f (z)   f (z)  
j
j
j
j m j 0 j 0



for all z  D and n > m > N. Thus,
n m 1  m 1


lim  f (z)   f (z)   f (z)   f (z)  
n j 0 j j 0 j j 0 j j 0 j




 n  
for m > N. (The limit of a series  a j  is almost always written as  a .)
  j
 j 0  j 0


8.3 Power Series
We are particularly interested in series of functions in which the partial sums are polynomials
of increasing degree:
s (z) = c + c (z – z ) + c (z – z ) + ... + c (z – z ) .
n
2
2
0
0
n
n
0
1
0
(We start with n = 0 for esthetic reasons.) These are the so-called power series. Thus,
 n 
j
a power series is a series of functions of the form  c (z z ) .

0 
 j 
 j 0 

Let’s look first at a very special power series, the so-called Geometric series:
 n j 
 z .  

 j 0 

Here,
s = 1 + z + z + ... + z , and
n
2
n
zs = z + z + z + ... + z .
n+1
3
2
n
Subtracting the second of these from the first gives us
(1 – z)s = 1 – z .
n+1
n
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