Math 411: Honours Complex Variables - University of Alberta

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

58 CHAPTER 9. HOLOMORPHIC FUNCTIONS ON ANNULI Split∂BP(z0)and∂Bρ(z0) − intofinitelymanyarcsegmentsα1,...,αn andβ1,...,βn, respectively, andconnectthemwithlinesegmentsγ1,...,γn asshownbelowforn = 3: We thus obtain � ∂BP(z0) α3 � f(ζ)dζ + γ − 3 γ3 β3 ∂Bρ(z0) − γ1 γ − 1 β2 α2 f(ζ)dζ = = n� � j=1 n� � j=1 β1 αj γ − 2 α1 γ2 f(ζ)dζ + n� � j=1 βj αj⊕γ − j ⊕βj⊕γ (j+1)mod n f(ζ)dζ f(ζ)dζ Bymakingthearcsegmentsα1,...,αn andβ1,...,βn sufficiently small, wecanensure that each of the closed curves α1 ⊕γ − 1 ⊕β1 ⊕γ2,...,αn−1 ⊕γ − n−1 ⊕βn−1 ⊕γn,αn ⊕ γ − n ⊕βn ⊕γ1 lies inside a star-shaped open subset of Ar,R(z0):
It follows that � � f(ζ)dζ + ∂BP(z0) as claimed. ∂Bρ(z0) − f(ζ)dζ = n� � j=1 αj⊕γ − j ⊕βj⊕γ (j+1)mod n f(ζ)dζ = 0 Theorem 9.2 (Laurent Decomposition). Let z0 ∈ C, let r,R ∈ [0,∞] be such that r < R, and let f : Ar,R(z0) → C be holomorphic. Then there exists a holomorphic function g: BR(z0) → C and h: C\Br[z0] → C with f = g +h on Ar,R(z0). Moreover, h can be chosen such that lim h(z) = 0, in |z|→∞ which case g and h are uniquely determined. Proof. We prove the uniqueness assertion first. Let g1,g2 : BR(z0) → C and h1,h2 : C \ Br[z0] → C be holomorphic such that lim hj(z) = 0 for j = 1,2 and |z|→∞ f = g1 +h1 = g2 +h2. It follows that g1 −g2 = h2 −h1 on Ar,R(z0). Define F : C → C, z ↦→ � g1(z)−g2(z), z ∈ BR(z0), h2(z)−h1(z), z ∈ C\Br[z0]. 59
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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 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: Chapter 9 Holomorphic Functions on
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))
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
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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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