Review Exam III
Complex Analysis
Underlined Definitions: May be asked for on exam
Underlined Propositions or Theorems: Proofs may be asked for on exam
Underlined Homework Exercises: Problems may be asked for on exam
Double Underlined Homework Exercises: Similar problems will be asked for on exam
Double Underlined Named Theorems/Results: Statements may be asked for on exam
Chapter 7.2
Homework
7.2 Page 154 4, 6, 8, 10, 13
Definition Let G be a region. A (G )  
Theorem Let G be a region. Let { f n } A (G ) and let
f C (G, ) . If f n  f , then
f A (G ) and f n( k )  f ( k ) for each k  1 .
Hurwitz's Theorem
Corollary Let G be a region. Let { f n }  A (G ) and
f A (G ) be such that f n  f . If each f n is
non-vanishing on G , then either f is non-vanishing on G or else f  0 .
Definition A set F  A (G ) is locally bounded if . . .
Lemma A set F  A (G ) is locally bounded if and only if for each K  G there exists a constant M such
that | f ( z ) |  M for all f F
and for all z  K .
Montel's Theorem
Chapter 7.4
Homework
7.4 Page 163
4 , 5 , 6, 7
Definition A region G1 is conformally equivalent to a region G2 if . . .
Riemann Mapping Theorem
Chapter 7.5
Homework
Definition Let
7.5 Page 173 4, 5,
6 , 7, 9
{zn }   . Then, the infinite product

z
n 1
n


z
Proposition Let Re zn  0 for all n . Then, the product
n 1
n
converges to a non-zero number if and only

 log z
if the series
n 1
n
converges.
Proposition Let Re zn  0 for all n . Then, the series

z
series
n 1
n
n 1
n 1
n
converges absolutely if and only if the

z
n 1
Corollary Let Re zn  0 for all n . Then, the product
z
 log z
 1 converges absolutely.
Definition Let Re zn  0 for all n . The product


n
converges absolutely if . . .

z
n 1
n
converges absolutely if and only if series
 1 converges absolutely.
n
Theorem Let G be a region. Let { f n } A (G ) be such that no f n is identically 0. If
 [ f ( z )  1] converges in A (G) , then  f ( z ) converges in A
 f ( z ) is a zero of one or more of the factors f ( z ) .
n
n
(G ) . Further, each zero of
n
n
Definition An elementary factor E p ( z )  
Lemma If | z |  1 , then | E p ( z )  1|  | z |
p 1
Theorem Let {an }   be such that lim | an |  , an  0 for all n. If { pn } is a sequence of integers such
n
 r 



n  1  | an | 

that
for all
r  0 , then


E
n 1
pn
( z an ) converges to an entire function whose zero set is precisely {an } .
Furthermore, (*) is always satisfied if pn  n  1 .
Weierstrass Factorization Theorem
Chapter 7.6
pn 1
(*)
 z2 
Theorem. sin  z   z   1  2  .
n 
n 1 

Chapter 7.7
Homework
7.7 Page 185 1, 2, 3, 7, 8
Definition The gamma function ( z )  
Gauss's Formula
Gauss’s Functional Equation For z  0,  1,  2,  , ( z  1)  z( z ) .
Bohr-Mollerup Theorem

Integral Representation For Re z  0 , ( z ) 
e
 t z 1
t
dt .
0
 z n 
z
Lemma  1    converges to e in A (G )
 n  
Chapter 7.8
Definition The Riemann zeta function
 ( z)  

Integral Representation 1. For Re z  1 ,
 ( z ) ( z ) 

0
1 z 1
t dt .
et  1
1
1
1
 1
  t z 1 dt 

Extension 1. For Re z  0 ,  ( z )( z )    t
e
t
z
1
1




0

Integral Representation 2. For
0  Re z  1 ,  ( z )( z ) 

e
1
t
1 z 1
t dt
1
1
 1
  t z 1 dt .
t
1 t 
  e
0
Extension 2. For 1  Re z  1 ,
1
1 1
1
 1
 ( z ) ( z )    t
   t z 1 dt 

e
t
z
1
2
2



0

1  z 1
 1

1  et  1 t  t dt

Integral Representation 3. For 1  Re z  0 ,
 ( z ) ( z ) 
1 1  z 1
 1

0  et  1 t  2  t dt
Riemann's Functional Equation
Theorem
 ( z ) A ( \{1})
with a simple pole at z = 1 with residue 1. Outside of the strip
0  Re z  1 ,
 ( z ) is non-vanishing except for simple zeros at z  2,  4,  6,  .
Riemann Hypothesis
Euler’s Theorem For Re z  0 ,
numbers.


1
z
n  1  1  pn
 ( z)   

 , where { pn } is an enumeration of the prime
