Page 88 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 8: Series

Notes



<    , 
3 3 3
Now suppose, we have a sequence (f ) of continuous functions which converges uniformly on a
n
 
n 
n 
contour C to the function f. Then the sequence   f (z)dz  converges to f (z)dz. This is easy to

C   C

see. Let  > 0. Now let N be so that |f (z) – f(z)| < A for n > N, where A is the length of C. Then,
n
n 

f (z)dz   f(z)dz =  (f (z) f(z))dz
n
C C C
<  A  
A
whenever n > N.
Now suppose (f ) is a sequence of functions each analytic on some region D, and suppose the
n
sequence converges uniformly on D to the function f. Then f is analytic. This result is in marked
contrast to what happens with real functions—examples of uniformly convergent sequences of
differentiable functions with a non-differentiable limit abound in the real case. To see that this
uniform limit is analytic, let z  D, and let S = {z : |z – z | < r}  D. Now consider any simple
0
0
0
closed curve C  S. Each f is analytic, and so f (z)dz  for every n. From the uniform
n 
n
C
 

n 
convergence of (f ), we know that f(z)dz is the limit of the sequence   f (z)dz ,  and so

n
C C  
 f(z)dz  0. Morera’s theorem now tells us that f is analytic on S, and hence at z . Truly a miracle.
0
C
8.2 Series
A series is simply a sequence (s ) in which s = a + a + ... + a . In other words, there is sequence
2
1
n
n
n
(a ) so that s = s n – 1 + a . The s are usually called the partial sums. Recall from Mrs. Turner’s class
n
n
n
n
 n 
that if the series  a j  has a limit, then it must be true that lim(a ) 0.
  n n
 j 1 

 n 
Consider a series  fj(z) of functions. Chances are this series will converge for some values
 
 j 1 

of z and not converge for others. A useful result is the celebrated Weierstrass M-test: Suppose
(M) is a sequence of real numbers such that M  0 for all j > J, where J is some number., and
j
j
 n 
suppose also that the series  M j  converges. If for all z  D, we have |fj(z)|  M for all j > J,
  j
 j 1 

 n 
then the series  f (z) converges uniformly on D.
 j 
 j 1 

LOVELY PROFESSIONAL UNIVERSITY 81