Math 411: Honours Complex Variables - University of Alberta

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

56 CHAPTER 8. THE SINGULARITIES OF A HOLOMORPHIC FUNCTION Case 2: ˜g(z0) = 0. For z ∈ Bǫ0(z0), we have � � |f(z)|= � 1 �˜g(z) +w0 � � � 1 � ≥ |˜g(z)| −|w0| → ∞. z→z0 Hence, z0 is a pole of f, again contradicting the fact that z0 is an essential singularity. Problem 8.1. Let D ⊂ C be open, let f : D → C be holomorphic, and let z0 ∈ D. Show that the following are equivalent for n ∈ N: (i) f (k) (z0) = 0 for k = 0,...,n−1 and f (n) (z0) �= 0; (ii) there exists a holomorphic function g: D → C with g(z0) �= 0 such that f(z) = (z −z0) n g(z) for z ∈ D. If either condition holds, we say that z0 is a zero of f of order n. Problem 8.2. Let D ⊂ C be open, let f,g: D → C be holomorphic, and let z0 ∈ D be a zero of order n for f and of order m ≥ 1 for g. Show the singularity z0 of f g is (i) removable if m ≤ n and (ii) a pole of order m−n otherwise. Problem 8.3. Let D ⊂ C be open, let f: D → C be holomorphic, and let z0 ∈ C\D be an isolated singularity of f. (a) Show that, if z0 is a pole of order k of f, then it is a pole of order k+1 of f ′ . (b) Show that exp◦f has either a removable or an essential singularity at z0.
Chapter 9 Holomorphic Functions on Annuli Definition. Let z0 ∈ C, and let r,R ∈ [0,∞] be such that r < R. Then the annulus centered at z0 with inner radius r and outer radius R is defined as Ar,R(z0) := {z ∈ C : r < |z −z0|< R}. Theorem 9.1 (Cauchy’s Integral Theorem for Annuli). Let z0 ∈ C, let r,ρ,P,R ∈ [0,∞] be such that r < ρ of. The claim is equivalent to � � f(ζ)dζ + Consider ∂BP(z0) z0 r ∂Bρ(z0) − 57 f(ζ)dζ = 0. ρ P R
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Math 411: Honours Complex Variables
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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 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: Define g: C → C, z ↦→ ∞�
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 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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