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 7.1
Homework
7.1 Page 150 1, 2, 4, 5, 7, 8
Definition. Let G be a region and let (Ω , d ) be a complete metric space. Then, C (G , Ω ) = . . .
Proposition. Let G be a region. Then there exists a sequence of subsets {K n } of G such that
(i)
K n ⊂⊂ G
(ii)
K n ⊂ int( K n +1 )
∞
(iii)
∪K
n
=G
n =1
(iv)
K ⊂⊂ G implies K ⊂ K n for some n ∈
Lemma If ( S , d ) is a metric space, then ( S , μ ) is a metric space, where
μ ( s, t ) =
d ( s, t )
. A set O is
1 + d ( s, t )
open in ( S , d ) if and only if O is open in ( S , μ ) .
Definition. For K ⊂⊂ G and f , g ∈C (G , Ω ) , let
ρ K ( f , g ) = sup d ( f ( z ), g ( z )) ,
z∈K
σ K ( f , g) =
ρK ( f , g )
, Bρ ( f , δ ) = { g : ρ K ( f , g ) < δ } .
1 + ρK ( f , g )
K
Definition. For {K n } a compact exhaustion of a region G and for f , g ∈C (G , Ω ) let
ρ( f , g) = . . .
Proposition. (C (G , Ω ), ρ ) is a metric space.
Lemma 1.7
ε > 0 there exists δ > 0 and K ⊂⊂ G such that for f , g ∈C (G , Ω )
ρK ( f , g ) < δ ⇒ ρ ( f , g ) < ε
(ii) Given δ > 0 and there exists ε > 0 such that
ρ ( f , g ) < ε ⇒ ρK ( f , g ) < δ
(i) Given
Lemma 1.10 (i) A set O ⊂ C (G , Ω ) is open if and only if for each f ∈ O there exists
δ > 0 and
K ⊂⊂ G such that O ⊃ Bρ K ( f , δ )
(ii) A sequence { f n } ⊂ C (G , Ω ) converges to f (in the ρ metric) if and only if for each
K ⊂⊂ G { f n } converges to f in the ρ K metric.
Proposition (C (G , Ω ), ρ ) is a complete metric space.
Definition. A set F ⊂ C (G , Ω ) is normal . . .
Proposition. A set F ⊂ C (G , Ω ) is normal if and only if F is compact.
Proposition. A set F ⊂ C (G , Ω ) is normal if and only if for each
δ > 0 and K ⊂⊂ G there exist functions
n
f1 , f 2 ,
, f n ∈ F such that F ⊂ ∪ Bρ K ( f k , δ ) .
k =1
Definition. A set F ⊂ C (G , Ω ) is equicontinuous at a point z 0 ∈ G if . . .
Definition. A set F ⊂ C (G , Ω ) is equicontinuous on a set E ⊂ G if . . .
Proposition. Suppose a set F ⊂ C (G , Ω ) is equicontinuous at each point of G. Then, F is equicontinuous
on each K ⊂⊂ G .
Arzela-Ascoli Theorem
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
7.4 Page 163 4 , 5 , 6, 7
Homework
Definition A region G1 is conformally equivalent to a region G2 if . . .
Riemann Mapping Theorem
Chapter 7.5
7.5 Page 173 4, 5, 6 , 7, 9
Homework
Definition Let { z n } ⊂
∞
. Then, the infinite product
∏z
n =1
=
n
∞
∏z
Proposition Let Re z n > 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 z n > 0 for all n . Then, the series
∞
series
∑z
n =1
n
∞
∑ log z
n =1
∞
∏z
n =1
Corollary Let Re z n > 0 for all n . Then, the product
n
converges absolutely if . . .
∞
∏z
n =1
∑z
n =1
n
converges absolutely if and only if the
− 1 converges absolutely.
Definition Let Re z n > 0 for all n . The product
∞
n
− 1 converges absolutely.
n
converges absolutely if and only if series
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
pn +1
<∞
∞
∏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
Theorem. sin π z = π z
⎛
z2
1
−
⎜
∏
n2
n =1 ⎝
∞
⎞
⎟.
⎠
Chapter 7.7
Homework
7.7 Page 185 1, 2, 3, 7, 8
Definition The gamma function Γ ( z ) =
Gauss's Formula
Gauss’s Functional Equation For z ≠ 0, − 1, − 2,
, Γ ( z + 1) = z Γ ( z ) .
Bohr-Mollerup Theorem
∞
Integral Representation For Re z > 0 , Γ ( z ) =
∫e
0
⎧⎪⎛
⎪⎩⎝
Lemma ⎨⎜ 1 +
z⎞
⎟
n⎠
n
⎫⎪
z
⎬ converges to e in A (G )
⎪⎭
− t z −1
t
dt .
(*)
Chapter 7.8
Definition The Riemann zeta function
ζ ( z) =
∞
Integral Representation 1. For Re z > 1 ,
ζ ( z )Γ ( z ) =
∫e
0
t
1 z −1
t dt .
−1
1
1⎞
1
⎛ 1
Extension 1. For Re z > 0 , ζ ( z )Γ ( z ) = ∫ ⎜ t
− ⎟ t z −1 dt +
+
1
1
e
t
z
−
−
⎝
⎠
0
∞
∫
1
1 z −1
t dt
e −1
t
∞
Integral Representation 2. For 0 < Re z < 1 ,
ζ ( z )Γ ( z ) =
1 ⎞ z −1
⎛ 1
−
∫0 ⎜⎝ et − 1 t ⎟⎠ t dt .
Extension 2. For −1 < Re z < 1 ,
1
1 1⎞
1
⎛ 1
ζ ( z )Γ ( z ) = ∫ ⎜ t
− + ⎟ t z −1 dt −
+
1
2
2
e
t
z
−
⎝
⎠
0
∞
1⎞
⎛ 1
− ⎟ t z −1 dt
t
−1 t ⎠
∫ ⎜⎝ e
1
∞
Integral Representation 3. For −1 < Re z < 0 ,
ζ ( z )Γ ( z ) =
1 1⎞
⎛ 1
− + ⎟ t z −1 dt
t
−1 t 2 ⎠
∫ ⎜⎝ e
0
Riemann's Functional Equation
Theorem
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,
.
Riemann Hypothesis
Euler’s Theorem For Re z > 0 ,
numbers.
∞
⎛
1
−z
n = 1 ⎝ 1 − pn
ζ ( z) = ∏ ⎜
⎞
⎟ , where { pn } is an enumeration of the prime
⎠