Page 94 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 8: Series

Notes
 n 
number, and suppose also that the series  M j  converges. If for all z  D, we have
 
 j 1 

 n 
|fj(z)|  M for all j > J, then the series  f (z) converges uniformly on D.
j
j
  j 1  

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
0
n
1
0
0
0
n
2
(We start with n = 0 for esthetic reasons.) These are the so-called power series. Thus, a
 n 
j
power series is a series of functions of the form  c (z z ) .

0 
 j 
 j 0 

8.7 Keywords
Sequence: A sequence of complex numbers is a function g : Z  C from the positive integers into
+
the complex numbers.
Partial sums: A series is simply a sequence (s ) in which s = a + a + ... + a . In other words, there
1
2
n
n
n
is sequence (a ) so that s = s n – 1 + a . The s are usually called the partial sums.
n
n
n
n
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
0
n
0
1
0
n
0
2
(We start with n = 0 for esthetic reasons.) These are the so-called power series.
8.8 Self Assessment
1. A .................. of complex numbers is a function g : Z  C from the positive integers into
+
the complex numbers.
2. A necessary and sufficient condition for the convergence of a sequence (a ) is the celebrated
n
Cauchy criterion: given  > 0, there is an integer N so that .................. whenever n, m > N .
 
3. A sequence (f ) of functions on a domain D is the obvious thing: a function from the
n
positive integers into the set of .................. on D.
4. A series is simply a sequence (s ) in which s = a + a + ... + a . In other words, there is
2
n
n
n
1
sequence (a ) so that s = s + a . The s are usually called the ..................
n n n – 1 n n
5. 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
0
1
0
0
2
n
n
0
(We start with n = 0 for esthetic reasons.) These are the so-called ..................
6. if C is any contour inside the circle of convergence, and the function g is continuous on C,
then ..................
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