Complex Analysis
Underlined Definitions: May be asked for on exam
Underlined Propositions or Theorems: Proofs may be asked for on exam
Cauchy’s Integral Formula (#0 ). Let G be a region in
£
, let positively. Let f
∈ A
B ( a , r ). Let f
∈ A
. Then, for z
∈
( , ) ,
Cauchy’s Theorem (#0) . Let G be a region in
£
, let
. Then,
∫
γ f
=
0 .
( , )
⊂
G and let
γ be the circle C ( a , r ), oriented
=
1
2
π i
∫
γ w
− z
( , )
⊂
G dw .
and let
γ be a closed rectifiable curve in
Chapter 4.5
Definition.
γ ≈
0 for a closed rectifiable curve
γ
lying in a region G if . . .
Lemma 5.1.
Let
γ be a rectifiable curve and let ϕ be continuous on {
γ
}. For each m
≥
1 , let f ( ) m
=
γ
∫ ϕ
( w
− z
) m dw for z
∈ £
. Then, f m
∈ A
(
£
\{ }) and f
' m
( )
= m f m
+
1
( ) .
Cauchy’s Integral Formula (#1). Let G be a region in
£ and let f
∈ A in G such that
γ ≈
0 . Then, for z
∈
G \ { } , n
γ
. Let
γ be a closed rectifiable curve
=
1
2
π i
∫
γ w
− z dw .
Cauchy’s Integral Formula (#2). Let G be a region in
£ and let f
∈ A rectifiable curves in G such that
γ
1
+ γ
2
+ L + γ m
≈
0 . Then, for z
∈
G \
γ k m U
=
1 k m ∑
=
1 n (
γ k
; ) ( )
= k m ∑ 1 ∫
=
1
2
π i
γ k w
− z dw .
,
. Let ,
2
,
L
,
γ m be a closed
Cauchy’s Theorem (#1).
Let G be a region in
£ and let f
∈ A curves in G such that
γ
1
+ γ
2
+ L + γ m
≈
0 . Then, m ∑ ∫ k
=
1 γ k f
=
0
. Let ,
2
,
L
,
γ m be a closed rectifiable
Corollary.
Let G be a region in
£ and let f
∈ A
. Let
γ be a closed rectifiable curve in G such that
γ ≈
0 .
Then, for z
∈
G \ { } , n
γ
( )
= k
2
π
!
i
∫
γ
( w
− z
) k
+
1
Morera’s Theorem Let G be a region in
£ and let f
∈ C dw .
. Suppose that
∫
T f
=
0 for every triangular path
T
⊂
G . Then, f
∈ A
.

Chapter 4.6
Definition.
γ
0
~
γ
1
for two closed rectifiable paths
γ γ
1
lying in a region G if . . .
Proposition.
“~” is an equivalence relation
Examples of pairs of curves ,
1
(and the overlying region G ) which are homotopic and of pairs of curves ,
1
(and the overlying region G ) which are not homotopic
Problems about constructing the homotopy function for a pair of curves
γ γ
1
(and the overlying region G ) which are homotopic
Definition. A set S is convex if . . .
Definition. A set S is starlike (w.r.t. a ) if . . .
Examples of sets which are convex / starlike and of sets which are not convex / starlike
Cauchy’s Theorem (#2)
γ ≈
0 .
. Let G be a region in
£ and let
γ be a closed rectifiable curve in G . If
γ
~ 0 , then
Corollary . Let G be a region in
£ which is starlike and let
Then,
∫
γ f
=
0 .
Cauchy’s Theorem (#3) . Let G be a region in
£
and let f f
∈ A
∈ A
G such that
γ
0
~
γ
1
. Then,
0
γ
∫ f
= ∫
γ
1 f .
Corollary.
Let G be a region in
£ and let f
∈ A
. Let
γ be a closed rectifiable curve in G .
. Let Let
γ
0
,
γ
1 be closed rectifiable curves in
. Let
γ be a closed rectifiable curve in G such that
γ
~ 0 .
Then,
∫
γ f
=
0 .
Definition.
γ
0
~
FEP
γ
1
for two rectifiable paths ,
1
lying in a region G if . . .
Examples of pairs of curves
γ γ
1 curves
γ γ
1
(and the overlying region G ) which are FEP-homotopic and of pairs of
(and the overlying region G ) which are not FEP-homotopic
Independence of Path Theorem.
Let G be a region in
£
and let
G such that γ
0
~
FEP
γ
1
. Then,
γ
0
∫ f
= ∫
γ
1 f .
f
∈ A
. Let Let
γ
0
,
γ
1 be rectifiable curves in

Definition. A region G is simply connected if . . .
Examples of regions which are simply connected and of regions which are not simply connected
Cauchy’s Theorem (#4) . Let G be a region in
£ which is simply connected and let closed rectifiable curve in G . Then,
∫
γ f
=
0 .
Corollary.
Let G be a region in
£ which is simply connected and let f
∈ A f
∈ A
. Let
γ be a
. Then, f has a primitive on G .
Definition. A function f has a branch-of-log f on a region G if . . .
Corollary.
Let G be a region in
£ which is simply connected and let branch-of-log f on G .
f
∈ A
. If ( )
≠
0 on G , then f has a
Chapter 4.7
Theorem 7.2.
Let G be a region in
£
and let f
∈ A
. Suppose Z f
= a a
1 2
,
L
, a m
} . Let
γ be a closed rectifiable curve in G such that
γ ≈
0 and such that Z f
Corollary 7.3.
Let G be a region in
£
and let f
∈ A
∩
{ }
= ∅
. Then
1
2
π
∫ '( ) dz i
γ
( )
. Suppose Z f
− α
=
= k m ∑
=
1 n ( ; a k
) a a
1 2
,
L
, a m
} . Let
γ be a
Z closed rectifiable curve in G such that
γ ≈
0 and such that Z f
− α
∩
{ }
= ∅ . Then
1
2
π i
∫
γ
( )
− α dz
= k m ∑
=
1 n ( ; a k
)
Proposition . Let G be a region in
£
and let
γ ≈
0 . Let
σ = f o γ
. Let f
∈ A
. Let
γ be a closed rectifiable curve in belong to the same component of
£
. Then,
G such that n
σ α = n
σ β f
∑
− α
P n ( ; z k
( ))
Z f
∑
− β
P n
γ z
β
Definition. A function f has a simple zero at z = a if . . .
Theorem 7.4.
Let G be a region in
£
, let zero of order m at z = a , then there exists
ε >
( )
= ζ has exactly m simple roots in
( , )
⊂
G
0 and
δ >
and let
0 f
∈ A
. Let
such that for each
ζ ∈
.
( )
= α
. If ( )
the equation
− α has a
Corollary.
Open Mapping Theorem
Chapter 4.8
Cauchy-Goursat Theorem . Let G be a region. If f
∈ D
, then f
∈ A
.

Chapter 5.1
Definition. A function f has an isolated singularity at z = a if . . .
Examples of functions which have isolated singularities
Definition. Removable Singularity, Pole, Essential Singularity
Examples of functions which have removable singularities, poles, essential singularities
Problems about classifying singularities as removable singularities, poles, essential singularities
Proposition . Let G be a region in
£ and let f have an isolated singularity at z = a , equivalent.
a
∈
G .
The following are a.
b.
f has a removable singularity at z = a f is bounded (in modulus) on a punctured neighborhood of a c.
d.
lim ( ) z
→ a exists lim ( z
→ a z
−
) ( )
=
0
Definition. A function f has a pole of order m at z = a if . . .
Proposition.
Let G be a region in
£ and let f have an isolated singularity at z = a .
f has a pole of order m at z = a if and only if there exists a function g analytic and non-vanishing on a neighborhood of z = a such that
=
( z
− a ) m
.
Definition. The singular part of a function f which has a pole of order m at z = a is . . .
Problems about identifying the singular parts of functions with poles
Definition. A double infinite series
∞ ∑ n
=−∞ z n
is absolutely convergent if . . .
Definition. A double infinite series
∞ ∑ n
=−∞ n
is uniformly convergent on a region S if . . .
Examples of double infinite series which are absolutely convergent / uniformly convergent.
Definition. ann a r r
1 2
=

Theorem (Laurent Series Expansion).
Let functions f
2
∈ A
B a R
2
)) and f
1
∈ A
( ) f
=
∈ A
( ( ;
1
,
2
)) such that f
= f
2
+ f
1
. Let G = on
£ ann a R R
B a R
1 2
)
1
) . Then there exist
. Furthermore, n
∞
∑
=
0 a n
( z
− a ) n on B a R
2
) (2) and
( )
= n
∞ ∑
=
1 a
− n
( z
− a )
− n on G (1) where a n
=
1
2
π i
∫
( w
− a ) n
+
1 dw , R
1
R
2
(*) and the coefficient a n
in (*) is independent of the choice of
ρ
. The series for f
2
in (2) converges absolutely on
B a R
2
) and uniformly on subsets B a r
2
of B a R
2
) , 0
< <
R
2
. The series for f
1
in (1) converges absolutely on G and uniformly on subsets ann a r
1
∞
of G .
Furthermore, the series representation
=
∞ ∑ n
=−∞ a n
( z
− a ) n on ( ;
1
,
2
) is unique.
Corollary.
Let G be a region in
£ and let f have an isolated singularity at z = a .
Let the Laurent Series expansion of f on a punctured neighborhood of a . Then, a.
z = a is removable singularity of f in and only if a n
= 0 for all n
≤ −
1 b.
c.
=
∞
∑ n
=−∞ a n
( z
− a ) n z = a is a pole of f of order m if and only if a
− m
≠
0 and a
− n
=
0 for all n m 1 z = a is an essential singularity if and only if a n
≠
0 for infinitely many negative n be
Problems constructing Laurent Series expansions.
Problems constructing Laurent Series expansions for rational functions.
Casorati-Weierstrass Theorem.
Let G be a region in
£ and let f have an isolated singularity at z = a .
If z = a is an essential singularity of f , then for every
δ >
0 , ( ( , )')
= £
.