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 11 The General Cauchy Integral Theorem Definition. Let D ⊂ C be open. We call a closed curve γ in D homologous to zero if ν(γ,z) = 0 for each z ∈ C\D. That is, the interior of γ is a subset of D. Definition. An open connected subset D of C is simply connected if every closed curve in D is homologous to zero. Equivalently, the interior of every closed curve in D is a subset of D. Theorem 11.1 (Cauchy’s Integral Formula). Let D ⊂ C be open, let f : D → C be holomorphic, and let γ be a closed curve in D that is homologous to zero. Then, for n ∈ N0 and z ∈ D \{γ}, we have ν(γ,z)f (n) (z) = n! � 2πi γ f(ζ) dζ. (ζ −z) n+1 Proof. It is enough to prove the claim for n = 0: for n ≥ 1, differentiate the integral with respect to z and use induction. Define g: D ×D → C, (w,z) ↦→ � f(w)−f(z) w−z , w �= z, f ′ (z), w = z. We claim that g is continuous. To see this, let (w0,z0) ∈ D × D. As g is clearly continuous at (w0,z0) if w0 �= z0, we need only show that g is continuous at (z0,z0). Given ǫ > 0, choose δ > 0 small enough that Bδ[z0] ⊂ D and |f ′ (z) −f ′ (z0)|< ǫ for all z ∈ Bδ[z0]. For (w,z) ∈ Bδ(z0)×Bδ(z0) we find: • if w = z: |g(w,z)−g(z0,z0)|= |f ′ (z)−f ′ (z0)|< ǫ; 68
• if w �= z: � � |g(w,z)−g(z0,z0)| = � f(w)−f(z) � −f w −z ′ � � (z0) � � � � � = � 1 �w−z Thus, g is continuous. Next, define [f [z,w] ′ (ζ)−f ′ (z0)]dζ � h0: D → C, z ↦→ γ � � � � g(ζ,z)dζ. 69 ≤ sup |f ζ∈{[z,w]} ′ (ζ)−f ′ (z0)|≤ ǫ. We claim that h0 is holomorphic. It is easy to see that h0 is continuous. To see that it is indeed holomorphic, we shall show that it satisfies the Morera condition. Let ∆ ⊂ D be a triangle. For fixed ζ ∈ {γ}, the function D → C, z ↦→ g(ζ,z) isholomorphicasaconsequenceofRiemann’sRemovabilityCondition. Goursat’s Lemma thus yields � g(ζ,z)dz = 0 ∂∆ for each ζ ∈ {γ}. As a consequence, we find � 0 = � = � = γ ∂∆ ∂∆ so that h0 is holomorphic as claimed. Define �� ∂∆ �� γ � g(ζ,z)dz dζ � g(ζ,z)dζ dz h0(z)dz, � h1: extγ → C, z ↦→ γ f(ζ) ζ −z dζ.
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Math 411: Honours Complex Variables
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Contents 1 The Complex Numbers 5 2
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Chapter 1 The Complex Numbers Defin
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Proof. (i): If z = x+iy, then ¯z =
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Chapter 2 Complex Differentiation D
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(i) there exists c ∈ C such that
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Example. Let Then f is totally diff
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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 and 28: That is, F ′ (z) = f(z). Theorem
Page 29 and 30: ∆ (4) ∆ (1) ∆ (2) As the line
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: Proposition 10.2 (Winding Numbers A
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 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
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Index C is a Field, 5 Biholomorphis
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INDEX 121 straight line segment, 25
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Math 411: Honours Complex Variables - University of Alberta

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