Math 411: Honours Complex Variables - University of Alberta

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

34 CHAPTER 5. CAUCHY’S INTEGRAL THEOREM AND FORMULA This is how the situation looks like: ∂B − ǫ z We connect the boundaries of Br(z0) and Bǫ(z) through line segments: Consider the following two closed curves γ1 and γ2: z0 z0 z
γ1 Then it is clear that � � 1 dζ + ζ −z γ1 γ2 z0 z � 1 dζ = ζ −z ∂Br(z0) z0 z � 1 dζ + ζ −z ∂Bǫ(z) − γ2 1 ζ −z dζ. The sketches also show that there exist star-shaped open set Dj ⊂ C \ {z} with {γj} ⊂ Dj for j = 1,2. Cauchy’s integral theorem for star-shaped domains thus yields that for j = 1,2, which proves the claim. � γj 1 dζ = 0 ζ −z Theorem 5.3 (Cauchy’s Integral Formula for Circles). Let D ⊂ C be open, let f: D → C be holomorphic, and let z0 ∈ D and r > 0 be such that Br[z0] ⊂ D. Then we have for all z ∈ Br(z0). f(z) = 1 � f(ζ) 2πi ∂Br(z0) ζ −z dζ Remark. A consequence of Theorem 5.3 is that the values of f on all of Br[z0] are pre-determined by those on ∂Br(z0)! Proof of Theorem 5.3. Since Br[z0] is compact, we may choose R > 0 be such that Br[z0] ⊂ BR(z0) ⊂ D. Let z ∈ Br(z0), and note that z is a center for the star-shaped domain BR(z0). Define g: D → C, ζ ↦→ � f(ζ)−f(z) ζ−z , ζ �= z, f ′ (z), ζ = z. 35
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 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: four subtriangles, denoted by ∆(z
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 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
Page 121:
INDEX 121 straight line segment, 25
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Math 411: Honours Complex Variables - University of Alberta

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