Math 411: Honours Complex Variables - University of Alberta

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

92CHAPTER13. FUNCTIONTHEORETICCONSEQUENCESOFTHERESIDUETHEOREM Proof. Suppose that f is not constant. Let z0 ∈ D be arbitrary, and define gn: D \{z0} → C, z ↦→ fn(z)−fn(z0) for n ∈ N. Then g1,g2,... have no zeros. Since f is not constant, the function D \{z0} → C, z ↦→ f(z)−f(z0) is not zero, so that it has no zeros by Hurwitz’s theorem, i.e. f(z) �= f(z0) for all z ∈ D, z �= z0. Theorem 13.5 (Rouché’s Theorem). Let D ⊂ C be open and simply connected, and let f,g : D → C be holomorphic. Suppose that γ is a closed curve in D such that intγ = {z ∈ D \{γ} : ν(γ,z) = 1} and that |f(ζ)−g(ζ)|< |f(ζ)| for ζ ∈ {γ}. Then f and g have the same number of zeros in intγ (counting multiplicity). Proof. For t ∈ [0,1], define ht := f +t(g−f), so that h0 = f and h1 = g. Also, since |t(g −f)|≤ |g −f|< |f| for any t ∈ [0,1] on {γ}, the functions ht have no zeros on {γ}. For t ∈ [0,1], let n(t) ∈ N0 denote the number of zeros of ht in intγ. Since the functions ht have no poles in D, we know from the Argument Principle that n(t) = 1 2πi � γ h ′ t(ζ) 1 dζ = ht(ζ) 2πi � γ f ′ (ζ)+t(g ′ (ζ)−f ′ (ζ)) f(ζ)+t(g(ζ)−f(ζ)) dζ. We thus see that n(t) is a continuous function of t. But since n(t) can only taken on integer values, it must be constant on [0,1]; in particular, n(0) = n(1). Example. How many zeros does z 4 −4z +2 have in D? Set g(z) := z 4 −4z +2 and f(z) = −4z +2. For ζ ∈ ∂D, we have |f(ζ)|≥ |−4z|−2 = 4−2 = 2, so that |f(ζ)−g(ζ)|= |ζ 4 |= 1 < 2 ≤ |f(ζ)|. Since f has precisely one zero in D, so does g. Corollary 13.5.1 (Fundamental Theorem of Algebra). Let p be a polynomial with n := degp ≥ 1. Then p has n zeros (counting multiplicity).
Proof. Let p(z) = anz n +···+a1z +a0 with an �= 0, and let g(z) := anzn � � � , so that lim � p(z)−g(z) � � |z|→∞� g(z) � = 0. Choose R > 0 such that � � �� p(z)−g(z) � � g(z) � < 1 for z ∈ C with |z|≥ R. Consequently, if z ∈ ∂BR(0), we have |p(z)−g(z)|< |g(z)|. By Rouché’s Theorem, p thus has as many zeros in BR(0) as g, namely n. Since p has at most n zeros, these are all of the zeros of p. 93
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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: � ∞�
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Figure 3.2: Surface plot of cos(z)
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It follows that {an(z−z0) n } ∞
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Problem 3.1. Show in Theorem 3.2 th
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Definition. The length of a piecewi
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1. Given a < b < c and two curves
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That is, F ′ (z) = f(z). Theorem
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∆ (4) ∆ (1) ∆ (2) As the line
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Proof. Let z0 ∈ D be a center for
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four subtriangles, denoted by ∆(z
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γ1 Then it is clear that � � 1
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Proof. There is no loss of generali
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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 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: Definition. Let D ⊂ C be open, an
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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