Math 411: Honours Complex Variables - University of Alberta

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$Math 527 A1 Homework 5 (Due Nov. 26 in Class) Proof. i. Set u = v(x ...$

Chapter 5 Cauchy’s Integral Theorem and Formula Definition. Let D ⊂ C be open. If f : D → C is complex differentiable at each z ∈ D, then we call f holomorphic (or analytic) on D. Let z1, z2, and z3 be three different points in C. They span a triangle ∆. Its boundary can be parametrized as a curve with counterclockwise orientation. We denote this curve by ∂∆. ∆ z1 z2 Theorem 5.1 (Goursat’sLemma). LetD ⊂ C be open, letf: D → C be holomorphic, and let ∆ ⊂ D be a triangle. Then we have � f(ζ)dζ = 0. ∂∆ Proof. First, we note that the result holds trivially whenever z1, z2, and z3 are colinear. Otherwise, we can split ∆ at its medians into four subtriangles ∆ (1) , ∆ (2) , ∆ (3) , and ∆ (4) as shown in the following figure: 28 z3
∆ (4) ∆ (1) ∆ (2) As the line segments in the interior of ∆ also occur as their reversed paths, we have � 4� � f = f, so that �� ∂∆ ∂∆ Choose j ∈ {1,2,3,4} such that � � � that �� f� � ≤ ∂∆ j=1 4� �� j=1 ∂∆ (j) ∂∆ (j) � � f� � . ∆ (3) 29 ∂∆ (j) f � � is largest, and set ∆1 := ∆ (j) . It follows � �� f� � ≤ 4� � also, note that ℓ(∂∆1) = 1 2 ℓ(∂∆). Repeat this argument with ∆1 in place of ∆, and obtain a triangle ∆2 ⊂ ∆1 with and �� so that �� ∂∆1 � � f� � ; ℓ(∂∆2) = 1 2 ℓ(∂∆1) = 1 4 ℓ(∂∆) ∂∆ ∂∆1 � �� f� � ≤ 4� � Inductively, we obtain triangles � �� f� � ≤ 4� � ∂∆1 ∂∆2 � � f� � , � �� f� � ≤ 16� � ∆ ⊃ ∆1 ⊃ ∆2 ⊃ ··· ∂∆2 � � f� � .
Page 1 and 2: Math 411: Honours Complex Variables
Page 3 and 4: Contents 1 The Complex Numbers 5 2
Page 5 and 6: Chapter 1 The Complex Numbers Defin
Page 7 and 8: Proof. (i): If z = x+iy, then ¯z =
Page 9 and 10: Chapter 2 Complex Differentiation D
Page 11 and 12: (i) there exists c ∈ C such that
Page 13 and 14: Example. Let Then f is totally diff
Page 15 and 16: holds over C. We obtain: � ∞�
Page 17 and 18: Figure 3.2: Surface plot of cos(z)
Page 19 and 20: It follows that {an(z−z0) n } ∞
Page 21 and 22: Problem 3.1. Show in Theorem 3.2 th
Page 23 and 24: Definition. The length of a piecewi
Page 25 and 26: 1. Given a < b < c and two curves
Page 27: That is, F ′ (z) = f(z). Theorem
Page 31 and 32: Proof. Let z0 ∈ D be a center for
Page 33 and 34: four subtriangles, denoted by ∆(z
Page 35 and 36: γ1 Then it is clear that � � 1
Page 37 and 38: Proof. There is no loss of generali
Page 39 and 40: Proof. Apply Theorem 5.3 and 5.4. E
Page 41 and 42: is entire. Since p is a nonconstant
Page 43 and 44: (ii) for each compact K ⊂ D, the
Page 45 and 46: Proof. Let ζ ∈ ∂Br(z0), and no
Page 47 and 48: holds for some a0,a1,a2,... ∈ C a
Page 49 and 50: Proof. Let z0 ∈ D be such that |f
Page 51 and 52: Theorem 7.5 (Biholomorphisms of D).
Page 53 and 54: Define ⎧ ⎪⎨ g: D ∪{z0} →
Page 55 and 56: Define g: C → C, z ↦→ ∞�
Page 57 and 58: Chapter 9 Holomorphic Functions on
Page 59 and 60: It follows that � � f(ζ)dζ +
Page 61 and 62: Definition. The function h in Theor
Page 63 and 64: For the converse, let g: Br(z0) →
Page 65 and 66: Chapter 10 The Winding Number of a
Page 67 and 68: Proposition 10.2 (Winding Numbers A
Page 69 and 70: • if w �= z: � � |g(w,z)−
Page 71 and 72: Corollary 11.2.1. Let D be an open,
Page 73 and 74: Examples. 1. Let f(z) = eiz z 2 +1
Page 75 and 76: 12.1. APPLICATIONS OF THE RESIDUE T
Page 77 and 78: 12.1. APPLICATIONS OF THE RESIDUE T
Page 79 and 80:
12.1. APPLICATIONS OF THE RESIDUE T
Page 81 and 82:
12.2. THE GAMMA FUNCTION 81 12.2 Th
Page 83 and 84:
12.2. THE GAMMA FUNCTION 83 Since
Page 85 and 86:
12.2. THE GAMMA FUNCTION 85 6 5 4 3
Page 87 and 88:
Chapter 13 Function Theoretic Conse
Page 89 and 90:
Lemma 13.1. Let D ⊂ C be open and
Page 91 and 92:
Definition. Let D ⊂ C be open, an
Page 93 and 94:
Proof. Let p(z) = anz n +···+a1z
Page 95 and 96:
for (x,y) ∈ R 2 \{(0,0)}. The par
Page 97 and 98:
Proof. Of course, only (ii) =⇒ (i
Page 99 and 100:
so that u(z) = � 2π u(re 0 iθ )
Page 101 and 102:
Set K := supζ∈∂D|f(ζ)|. For z
Page 103 and 104:
Chapter 15 Analytic Continuation al
Page 105 and 106:
z0 ∈ D, the germ of f at z0—den
Page 107 and 108:
Chapter 16 Montel’s Theorem Defin
Page 109 and 110:
Define g: D → RM as follows: for
Page 111 and 112:
and note that z−i 1+ z+i g(f(z))
Page 113 and 114:
Example. Let z be a complex number.
Page 115 and 116:
By the Open Mapping Theorem, there
Page 117 and 118:
Proof. (i) =⇒ (ii) is Corollary 1
Page 119 and 120:
Index C is a Field, 5 Biholomorphis
Page 121:
INDEX 121 straight line segment, 25
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Math 411: Honours Complex Variables - University of Alberta

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