Math 411: Honours Complex Variables - University of Alberta

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

24 CHAPTER 4. COMPLEX LINE INTEGRALS Definition. A curve γ: [a,b] → C is called closed if γ(a) = γ(b). Proposition 4.1. Let D ⊂ C be open, and let f : D → C be continuous with an antiderivative. Then � f = 0 holds for each closed, piecewise smooth curve γ in D. Example. Let z0 ∈ C, let r > 0, and let γ γ: [0,2π] → C, θ ↦→ re iθ +z0, i.e. γ is a counterclockwise-oriented circle centered at z0 with radius r. Let n ∈ Z, and consider � γ (ζ −z0) ndζ. For n �= −1, let n+1 (z −z0) F : C → C, z ↦→ , n+1 so that F ′ (z) = (z −z0) n for all z ∈ C. It follows that � On the other hand, we have Consequently, � has no antiderivative. γ (ζ −z0) −1 dζ = � 2π 0 γ (ζ −z0) n dζ = 0. rieiθ � 2π dθ = idt = 2πi. reiθ 0 C\{z0} → C, z ↦→ 1 z −z0 Recall the following definition from multivariable calculus: Definition. A subset D ⊂ C is called connected if there are no open sets U,V ⊂ C with • U ∩D �= ∅ �= V ∩D; • U ∪V ⊃ D; • U ∩V ⊂ C\D. In other words, there are no open sets U and V, each containing points of D, such that every point of D lies in exactly one of the sets U and V. Definition. Let D ⊂ C be open. A function f : D → C is called locally constant if, for each z0 ∈ D, there exists ǫ > 0 such that Bǫ(z0) ⊂ D and f is constant on Bǫ(z0). The following curve constructions will be useful in understanding the relation between locally constant functions and connectivity.
1. Given a of γ1 and γ2 is the curve � γ1(t), t ∈ [a,b], γ1 ⊕γ2: [a,c] → C, t ↦→ γ2(t), t ∈ [b,c]. If γ1 and γ2 are piecewise smooth, then so is γ1 ⊕γ2, and we have � γ1⊕γ2 � f = for each continuous f: {γ1}∪{γ2} → C. γ1 � f + 2. For any curve γ: [a,b] → C, the reversed curve is defined as γ − : [a,b] → C, t ↦→ γ(a+b−t). If γ is piecewise smooth, then so is γ− , and we have � � f = − f for each continuous f: {γ} → C. γ − 3. We denote the straight line segment {z0 +t(z −z0) : t ∈ [0,1]} by [z0,z]. Proposition 4.2 (Locally Constant vs. Connectivity). Let D ⊂ C be open. Then the following are equivalent: (i) D is connected; (ii) every locally constant function f: D → C is constant; (iii) for any z,w ∈ D, there exists a piecewise smooth curve γ: [a,b] → D such that γ(a) = z and γ(b) = w. Proof. (iii) =⇒ (ii): Let f: D → C be a locally constant function, and let z,w ∈ D. Let γ: [a,b] → D be a piecewise smooth curve with γ(a) = z and γ(b) = w. Since f is locally constant, the function γ γ2 [a,b] → C, t ↦→ f(γ(t)) is differentiable with zero derivative and therefore constant. It follows that f(z) = f(γ(a)) = f(γ(b)) = f(w). (ii) =⇒ (i): Suppose that D is not connected. Then there exist non-empty open sets U,V ⊂ C with U ∩V = ∅ and U ∪V = D. Define f: D → C, z ↦→ f � 0, z ∈ U, 1, z ∈ V. 25
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: Definition. The length of a piecewi
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 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:
Page 79 and 80:
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))
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Example. Let z be a complex number.
Page 115 and 116:
By the Open Mapping Theorem, there
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Proof. (i) =⇒ (ii) is Corollary 1
Page 119 and 120:
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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