- Page 1: Math 411: Honours Complex Variables
- 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 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).
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Define ⎧ ⎪⎨ g: D ∪{z0} →
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Define g: C → C, z ↦→ ∞�
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Chapter 9 Holomorphic Functions on
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It follows that � � f(ζ)dζ +
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Definition. The function h in Theor
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For the converse, let g: Br(z0) →
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Chapter 10 The Winding Number of a
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Proposition 10.2 (Winding Numbers A
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• if w �= z: � � |g(w,z)−
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Corollary 11.2.1. Let D be an open,
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Examples. 1. Let f(z) = eiz z 2 +1
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12.1. APPLICATIONS OF THE RESIDUE T
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12.1. APPLICATIONS OF THE RESIDUE T
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12.1. APPLICATIONS OF THE RESIDUE T
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12.2. THE GAMMA FUNCTION 81 12.2 Th
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12.2. THE GAMMA FUNCTION 83 Since
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12.2. THE GAMMA FUNCTION 85 6 5 4 3
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Chapter 13 Function Theoretic Conse
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Lemma 13.1. Let D ⊂ C be open and
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Definition. Let D ⊂ C be open, an
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Proof. Let p(z) = anz n +···+a1z
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for (x,y) ∈ R 2 \{(0,0)}. The par
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Proof. Of course, only (ii) =⇒ (i
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so that u(z) = � 2π u(re 0 iθ )
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Set K := supζ∈∂D|f(ζ)|. For z
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Chapter 15 Analytic Continuation al
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z0 ∈ D, the germ of f at z0—den
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Chapter 16 Montel’s Theorem Defin
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Define g: D → RM as follows: for
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and note that z−i 1+ z+i g(f(z))
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Example. Let z be a complex number.
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By the Open Mapping Theorem, there
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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