Page 93 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes and apply the result of the previous section to conclude that


j
 g(s)S(s)ds   c g(s)(s z ) ds,or
j

0

C j 0 C
1 S(s)  c 1 (s z ) j ds. Thus

0


2 i (s z) 2 ds   j 2 i (s z) 2




j 0
C
C


S '(z)   jc (z  z ) j 1 .

j
0
j 0

8.6 Summary
The basic definitions for complex sequences and series are essentially the same as for the

real case. A sequence of complex numbers is a function g : Z  C from the positive
+
integers into the complex numbers. It is traditional to use subscripts to indicate the values
of the function. Thus, we write g(n)  z and an explicit name for the sequence is seldom
n
used; we write simply (z ) to stand for the sequence g which is such that g(n) = z . For
n
n
example,   i   is the sequence g for which g(n) = i .
 n  n
The number L is a limit of the sequence (z ) if given an  > 0, there is an integer N such that
 n 
|zn – L| <  for all n  N . If L is a limit of (z ), we sometimes say that (z ) converges to L.
n
n

We frequently write lim (z ) = L. It is relatively easy to see that if the complex sequence
n
(z ) = (u + iv ) converges to L, then the two real sequences (u ) and (v ) each have a limit:
n
n
n
n
n
(u ) converges to ReL and (v ) converges to ImL. Conversely, if the two real sequences (u )
n
n
n
and (v ) each have a limit, then so also does the complex sequence (u + iv ). All the usual
n
n
n
nice properties of limits of sequences are, thus, true:
lim(z ± w ) = lim(z ) ± lim(w );
n
n
n
n
lim(z w ) = lim(z ) lim(wn); and
n
n
n
 z  lim(z )
lim  n  = n .
 w n  lim(w )
n
provided that lim(z ) and lim(w ) exist. (And in the last equation, we must, of course, insist
n
n
that lim(w )  0.).
n
A series is simply a sequence (s ) in which s = a + a + ... + a . In other words, there is
 n n 1 2 n
sequence (a ) so that s = s n – 1 + a . The s are usually called the partial sums. Recall from
n
n
n
n
 n 
Mrs. Turner’s class that if the series  a j  has a limit, then it must be true that lim(a ) 0.
n


 j 1  n

 n 
Consider a series  fj(z) of functions. Chances are this series will converge for some


 j 1 

values 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
j
j
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