Page 92 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 8: Series

The number R is called the radius of convergence of the series, and the set |z – z | = R is called Notes
0
the circle of convergence. Observe also that the limit of a power series is a function analytic
inside the circle of convergence (why?).

8.4 Integration of Power Series

Inside the circle of convergence, the limit


S(z)   c (z z ) j

0
j
j 0

is an analytic function. We shall show that this series may be integrated ”term-by-term”—that
is, the integral of the limit is the limit of the integrals. Specifically, if C is any contour inside the
circle of convergence, and the function g is continuous on C, then

j
 g(z)S(z)dz   c g(z)(z z ) dz.
j

0
C j 0 C

Let’s see why this. First, let  > 0. Let M be the maximum of |g(z)| on C and let L be the length
of C. Then there is an integer N so that
 
j
 c (z z ) 

j
0
j n ML

for all n > N. Thus,
  j  
   g(z)  c (z z ) dz  ML  , 

0 

j
C j n  ML

Hence,
n 1   

j
j
 g(z)S(z)dz   c g(z)(z z ) dz    g(z) c (z z ) dz   .
j


0 
j
0
C j 0 C C  j n 


8.5 Differentiation of Power Series
Again, let

S(z) =  c (z z )j.

0
j
j 0

Now we are ready to show that inside the circle of convergence,

j 1


S'(z)   jc (z z ) .
0
j
j 1

Let z be a point inside the circle of convergence and let C be a positive oriented circle centered
at z and inside the circle of convergence. Define
1
g(s)  ,
2 i(s z) 2


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