Review Part III
Complex Analysis
Underlined Definitions: May be asked for on exam
Underlined Propositions or Theorems: Proofs may be asked for on exam
Chapter 4.1
Riemann-Stieltjes Integrals
Definition of function of bounded variation and total variation
Proposition (1.3) If γ :[ a , b ] →
is piecewise smooth, then γ is of bounded variation and
b
V (γ ) = ∫ | γ '(t ) | dt .
a
Definition of Riemann-Stieltjes Integral
is continuous and if γ :[ a , b ] →
Theorem (1.4) If f :[ a , b ] →
b
Riemann-Stieltjes integral
is of bounded variation, then the
b
∫f
d γ = ∫ f (t ) d γ (t ) exists.
a
a
(Proof uses Cantor’s Theorem II.3.7).
Proposition 1.7 Let f , g :[ a , b ] →
α , β ∈ . Then,
b
be continuous, let γ , σ :[ a , b ] →
b
be of bounded variation and let
b
∫ (α f + β g )d γ = α ∫ f d γ + β ∫ gd γ
a)
a
b)
a
a
b
b
b
a
a
a
∫ fd (aγ + βσ ) = α ∫ f d γ + β ∫ gdσ
Proposition Let f :[ a , b ] →
be continuous and let γ :[ a , b ] →
b
a < t0 < t1 <
< t n = b , then
n
∫ fd γ = ∑ ∫
fd γ
k =1 tk −1
a
Theorem (1.9) If γ :[ a , b ] →
tk
be of bounded variation. If
b
is piecewise smooth and f :[ a , b ] →
, then
∫f
a
Definition for a path γ :[ a , b ] →
b
∫
a
a
∫
and length of {γ } = d γ . For γ piece-wise smooth, length
a
of {γ } = | γ '(t ) | dt .
d γ = ∫ f (t )γ '(t ) dt .
of trace of γ , {γ } .
Definition of rectifiable path γ :[ a , b ] →
b
b
Definition of line integral: Let γ :[ a , b ] →
b
define line integral
∫γ f = ∫ ( f
a
be rectifiable path and let f :{γ } →
be continuous,
b
γ ) d γ = ∫ f (γ (t )) d γ (t ) = ∫ f ( z ) dz
γ
a
∫γ
Note: if γ piece-wise smooth, then
b
b
b
a
a
a
f = ∫ ( f γ ) d γ = ∫ f (γ (t )) d γ (t ) = ∫ f (γ (t ))γ '(t ) dt = ∫ f ( z ) dz
γ
Problems about computing line integrals using the definition
Definition of a change of parameter ϕ
Proposition If ϕ is a change of parameter, i.e., if ϕ :[c , d ] → [ a , b ] , ϕ is continuous, strictly increasing
and ϕ is onto, then for γ :[ a , b ] →
Definition:
continuous, then
∫γ f
=
∫γ ϕ f
(1) a curve as an equivalance class of rectifiable paths;
(2) the trace of a curve is the trace of a representative;
(3) a curve is smooth if some representative is smooth;
(4) a curve is closed if the initial and terminal points on the trace are the same.
Definition for γ :[ a , b ] →
∫γ
a rectifiable path and f :{γ } →
a rectifiable path of −γ and of | γ (t ) | and definition
b
f ( z ) | dz | = ∫ f (γ (t )) d | γ | (t )
a
Proposition (1.17) Let γ :[ a , b ] →
a)
∫
−γ
b)
be a rectifiable path and let f :{γ } →
be continuous. Then,
f = −∫ f
γ
| ∫ f |≤ ∫ | f | | dz |≤ max | f ( z ) | V (γ )
γ
γ
z∈{γ }
Theorem (Fundamental of Theorem of Calculus for Line Integrals) Let G be a region and let γ be a
rectifiable path in G with initial and terminal points α and β , resp. If f : G → is continuous and if
f has a primitive on G, say F, then
∫γ f
β
= F ( z) α .
Corollary Let G be a region and let γ be a closed rectifiable path in G .
and if f has a primitive on G, say F, then
∫γ f
If f : G →
is continuous
=0.
Problems about computing line integrals using the Fund. Thm. of Calc. for Line Integrals
Chapter 4.2
Proposition 2.1 (Leibnitz’s Rule)
Integrals
a)
∫
( w − z ) n dw = 0, n = 0,1, 2, 3,
| w| =1
b)
⎧ n = 2, 3, 4, 5,
dw
=
0,
⎨
∫ (w − z )n
| z |≠ 1
⎩
| w| =1
c)
⎧ 0, | z |> 1
dw
=⎨
w − z ⎩ 2π i, | z |< 1
| w| =1
∫
Cauchy Integral Formula #0 Let f : G →
z ∈ B (a, r ) , f ( z) =
be analytic and suppose that B ( a , r ) ⊂ G . For
1
f ( w)
dw
∫
2π i |w − a|= r w − z
Problems about computing line integrals using the CIF #0
Lemma (2.7) Let γ be a rectifiable curve. Suppose that Fn and F are continuous on {γ } and that
{Fn } converges uniformly on {γ } to F. Then, lim ∫ Fn = ∫ lim Fn = ∫ F
n →∞
Theorem 2.8 Let G be a region and let f : G →
series representation on B ( a , R ) , say
γ
γ
n →∞
γ
be analytic. Let B ( a , R ) ⊂ G . Then, f has a power
∞
f ( z ) = ∑ an ( z − a ) n
(1)
n=0
where the coefficients an =
f ( n ) (a )
1
f ( w)
dw , for any choice 0 <
=
∫
n!
2π i | w − a| = ρ ( w − a ) n +1
the radius of convergence of the power series (1) is at least R.
ρ < R . Furthermore,
Corollaries (Hypothesis: Let G be a region and let f : G →
a)
b)
be analytic. Let B ( a , R ) ⊂ G . )
the radius of convergence of the power series (1) is equal to dist( a , ∂G ) , i.e., the
distance (from a) to the nearest singularity of f
f ( n ) (a ) =
n!
f ( w)
dw
∫
2π i |w− a|= ρ ( w − a ) n +1
c)
Cauchy’s Estimate If | f ( z ) |≤ M on B ( a , R ) , then | f ( n ) ( a ) |≤
d)
f has a primitive on B ( a , R ) , namely F ( z ) =
∞
an
∑ n + 1( z − a )
n!M
.
Rn
n +1
n=0
e)
Proposition 2.15 Suppose γ is a closed rectifiable curve in B ( a , R ) . Then,
∫γ f
=0
Chapter 4.3
Division Algorthim
Definition: Let G be a region and let f : G →
of order m (multiplicity m) at z = a if
be analytic and let f (a) = 0. We say that f has a zero
a) there exists g ∈ A (G ) such that (i) f ( z ) = ( z − a ) m g ( z ) and (ii) g ( a ) ≠ 0
or alternatively
b) (i) f ( a ) = f '( a ) = f ''( a ) =
= f ( m −1) ( a ) = 0 and f ( m ) ( a ) ≠ 0
Definition: entire function
Liouville’s Theorem If f is a bounded entire function, then f is constant.
Fundamental Theorem of Algebra Every non-constant polynomial with complex coefficients has a root
in
Corollary: Every polynomial with complex coefficients of degree n has exactly n roots (counted
according to multiplicity)
Identity Theorem: Let G be a region. Let f be analytic on G. TFAE
a)
f ≡0
b)
there exists a point a ∈ G such that f ( n ) ( a ) = 0, n = 0,1, 2,3,
c)
Z f = { z ∈ G : f ( z ) = 0} has a limit point in G.
Corollaries:
a)
Let g , h ∈ A (G ) and g ( z ) = h ( z ) for z ∈ S ⊂ G . If S has a limit point in G, then
g ≡h.
b)
Let G be a region. Let f be analytic on G. Suppose that f is not identically 0 on G. If
a ∈ G and f (a) = 0, then there exists an integer m such that f has a zero at z = a of
multiplicity m.
c)
Isolated Zeros. Let G be a region. Let f be analytic on G. Suppose that f is not
identically 0 on G. If a ∈ G and f (a) = 0, then there exists an R > 0 such that on
B ( a , R ) \ {a} we have f ( z ) ≠ 0 .
d)
Let G be a region. Let f be analytic on G. Suppose that f is not identically 0 on G.
Let K be a compact subset of G. Then, f has at most a finite number of zeros on K.
Further, f has at most a countable number of zeros on G.
Maximum Modulus Theorem Let G be a region. Let f be analytic on G. If there exists a point
a ∈ G such that | f ( a ) |≥| f ( z ) | for all z ∈ G , then f is constant on G.
Extension Let G be a region. Let f be analytic on G. If there exists a point a ∈ G and r > 0 such
that | f ( a ) |≥| f ( z ) | for all z ∈ B ( a , r ) , then f is constant on G.
Chapter 4.4
Proposition: Let γ be a closed rectifiable curve and let a ∉ {γ } . Then,
1
dz
is an integer.
∫
2π i γ z − a
Definition: Winding number of γ wrt to a (index of γ wrt to a) n(γ , a ) =
1
dz
, for γ a closed
∫
2π i γ z − a
rectifiable curve and a ∉ {γ } .
Interpretation: 2π n (γ , a ) represents the total change in arg(γ (t ) − a ) as γ (t ) parametrizes the curve
{γ } , i.e., n (γ , a ) represents the total number of times γ winds around a.
Theorem Let γ be a closed rectifiable curve. Then, n (γ , a ) is constant on components of
Also, n (γ , a ) = 0 on the unbounded component of \ {γ }
\ {γ } .
Cauchy’s Integral Formula (#0). Let G be a region in
, let B ( a , r ) ⊂ G and let γ be the circle
C(a,r), oriented positively. Let f ∈ A (G ) . Then, for z ∈ B ( a , r ) , f ( z ) =
Cauchy’s Theorem (#0).
∫γ f
f ( w)
dw .
w− z
∫γ
, let B ( a , r ) ⊂ G and let γ be a closed rectifiable
Let G be a region in
curve in B(a,r). Let f ∈ A (G ) . Then,
1
2π i
=0.
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 }.
fm ( z) = ∫
γ
ϕ ( w)
(w − z)
m
dw for z ∈
For each m ≥ 1 , let
\ {γ } . Then, f m ∈ A ( \ {γ }) and f m' ( z ) = m f m +1 ( z ) .
Cauchy’s Integral Formula (#1). Let G be a region in
and let f ∈ A (G ) . Let γ be a closed
rectifiable curve in G such that γ ≈ 0 . Then, for z ∈ G \ {γ } , n(γ ; z ) f ( z ) =
Cauchy’s Integral Formula (#2). Let G be a region in
rectifiable curves in G such that γ 1 + γ 2 +
m
m
1
k =1 2π i
∑ n(γ k ; z ) f ( z ) = ∑
k =1
∫
γ
k
1
2π i
∫γ
f ( w)
dw .
w− z
and let f ∈ A (G ) . Let γ 1 , γ 2 ,
, γ m be closed
+ γ m ≈ 0 . Then, for z ∈ G \ ∪ k =1{γ k } ,
n
f ( w)
dw .
w− z
and let f ∈ A (G ) . Let γ be a closed rectifiable curve
Cauchy’s Theorem (#1). Let G be a region in
in G such that γ ≈ 0 . Then,
∫γ f
=0
Cauchy’s Theorem (#2). Let G be a region in
and let f ∈ A (G ) . Let γ 1 , γ 2 ,
m
rectifiable curves in G such that γ 1 + γ 2 +
+ γ m ≈ 0 . Then,
∑∫ f
, γ m be a closed
=0
k =1 γ k
Corollary. Let G be a region in
and let f ∈ A (G ) . Let γ be a closed rectifiable curve in G such that
γ ≈ 0 . Then, for z ∈ G \ {γ } , n(γ ; z ) f ( k ) ( z ) =
Morera’s Theorem Let G be a region in
k!
2π i
f ( w)
∫γ ( w − z )
k +1
dw .
and let f ∈C (G ) . Suppose that
∫f
T
triangular path T ⊂ G . Then, f ∈ A (G ) .
= 0 for every