Page 87 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes
stand for the sequence g which is such that g(n) = z . For example,   i   is the sequence g for
n
 n 
i
which g(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. We
n
n

frequently write lim (z ) = L. It is relatively easy to see that if the complex sequence (z ) =
n
n
(u + iv ) converges to L, then the two real sequences (u ) and (v ) each have a limit: (u ) converges
n
n
n
n
n
to ReL and (v ) converges to ImL. Conversely, if the two real sequences (u ) and (v ) each have a
n
n
n
limit, then so also does the complex sequence (u + iv ). All the usual nice properties of limits of
n
n
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 that
n
n
lim(w )  0.)
n
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 |a – a | <  whenever n,m > N .
 n m 
A sequence (f ) of functions on a domain D is the obvious thing: a function from the positive
n
integers into the set of complex functions on D. Thus, for each z  D, we have an ordinary
sequence (fn(z)). If each of the sequences (fn(z)) converges, then we say the sequence of functions
(f ) converges to the function f defined by f(z) = lim(f (z)). This pretty obvious stuff. The sequence
n
n
(f ) is said to converge to f uniformly on a set S if given an  > 0, there is an integer N so that
n

|f (z) – f(z)| <  for all n  N and all z  S.
n

Notes It is possible for a sequence of continuous functions to have a limit function
that is not continuous. This cannot happen if the convergence is uniform.
To see this, suppose the sequence (f ) of continuous functions converges uniformly to f on a
n
domain D, let z  D, and let  > 0. We need to show there is a  so that |f(z ) – f(z)| <  whenever
0
0

|z – z| < . Let’s do it. First, choose N so that |f (z) – f(z)| < . We can do this because of the
3
0
N
uniform convergence of the sequence (f ). Next, choose  so that |f (z ) – f (z)| <  3 whenever
N
0
n
N
|z – z| < . This is possible because f is continuous.
0
N
Now then, when |z – z| < , we have
0
|f(z ) – f( )| = |f(z ) – f (z ) + f (z ) – f (z) + f (z) – f(z)|
z
0
N
N
0
N
0
0
N
 |f(z ) – f (z )| + |f (z ) – f (z)| + |f (z) – f(z)|
N
0
N
N
N
0
0
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