Review Exam III
Complex Analysis
Underlined Definitions: May be asked for on exam
Underlined Propositions or Theorems: Proofs may be asked for on exam
Double Underlined Named Theorems/Results: Statements may be asked for on exam
Chapter 7.2
Definition Let G be a region.
A (G ) =L
(G ) and let f ∈C (G ,£ ) . If f n → f , then
for each k ≥ 1 .
Theorem Let G be a region. Let { f n} ⊂ A
f ∈A (G ) and f n(k ) → f ( k )
Hurwitz's Theorem
(G ) and f ∈A (G ) be such that f n → f . If each f n is nonvanishing on G , then either f is non-vanishing on G or else f ≡ 0 .
Corollary Let G be a region. Let { f n} ⊂ A
Definition A set F
⊂ A (G ) is locally bounded if . . .
⊂ 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 .
Lemma A set F
Montel's Theorem
Chapter 7.4
Definition A region
G1 is conformally equivalent to a region G2 if . . .
Riemann Mapping Theorem
Chapter 7.5
Definition Let { zn} ⊂ £ . Then, the infinite product
∞
∏z
n=1
n
=L
∞
Proposition Let
Re zn > 0 for all n . Then, the product ∏ z n converges to a non-zero number if and only if
n=1
∞
the series
∑ log z
n=1
n
converges.
Proposition Let Re zn > 0 for all n . Then, the series
∞
∑ log z
n=1
∞
∑z
n
n
converges absolutely if and only if the series
− 1converges absolutely.
n =1
∞
Definition Let
Re zn > 0 for all n . The product ∏ z n converges absolutely if . . .
n=1
∞
Corollary Let
Re zn > 0 for all n . Then, the product ∏ z n converges absolutely if and only if series
n=1
∞
∑z
n
− 1converges absolutely.
n =1
Theorem Let G be a region. Let { f n} ⊂ A
in A
(G ) , then
factors f n ( z ) .
∏f
n
(G ) be such that no f n is identically 0. If
( z ) converges in A (G ) . Further, each zero of
∏f
n
∑[ f
n
( z ) − 1] converges
( z ) is a zero of one or more of the
Definition An elementary factor E p ( z ) = L
Lemma If
p +1
| z | ≤ 1, then | E p ( z) −1| ≤ | z |
pn +1
 r 
|= ∞ , an ≠ 0 for
<∞
Theorem Let {an } ⊂ £ be such that lim | an∑
all n. If { pn } is a sequence of integers such that
(*)
n→∞
|
a
|
n = 1
n 
∞
for all r > 0 , then
∞
∏E
n=1
is always satisfied if
pn
( z an ) converges to an entire function whose zero set is precisely {an } . Furthermore, (*)
pn = n −1 .
Weierstrass Factorization Theorem
Chapter 7.6
 z2 
Theorem. sin π z = π z ∏  1 − 2  .
n 
n =1 
∞
Chapter 7.7
Definition The gamma function
Gauss's Formula
Γ( z ) = L
Gauss’s Functional Equation For
z ≠ 0,− 1, − 2, L , Γ(z + 1) = zΓ( z) .
Bohr-Mollerup Theorem
∞
Integral Representation For Re z > 0 ,
Γ( 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
ζ ( z) = L
Integral Representation 1. For Re z > 1 ,
∞
ζ ( z )Γ( z) =
∫
0
1 z −1
t dt .
et − 1
∞
1
1
1
 1
−  t z −1 dt +
+
Extension 1. For Re z > 0 , ζ ( z )Γ( z) = ∫  t
e
−
1
t
z
−
1


0
∞
Integral Representation 2. For 0 < Re z < 1 ,
ζ ( z )Γ( z) =
 1
1
∫  e − 1 − t  t
∫
1
z −1
t
1 z −1
t dt
et −1
dt .
0
Extension 2. For −1 < Re z < 1 ,
1
1 1
1
 1
ζ ( z )Γ( z) = ∫  t
− +  t z −1 dt −
+
e
−
1
t
2
2
z


0
∞
1  z −1
 1
−
∫1  et − 1 t  t dt
∞
Integral Representation 3. For −1 < Re z < 0 ,
ζ ( z )Γ( z) =
1 1  z −1
 1
−
∫0  et − 1 t + 2  t dt
Riemann's Functional Equation
ζ (z ) ∈A (£ \{1}) with a simple pole at z = 1 with residue 1. Outside of the strip 0 ≤ Re z ≤ 1 ,
ζ ( z) is non-vanishing except for simple zeros at z = −2, − 4, − 6, L .
Theorem
Riemann Hypothesis
Euler’s Theorem For Re z > 0 ,
 1
ζ (z) = ∏
−z
n = 1  1 − pn
∞

 , where { pn } is an enumeration of the prime numbers.
