Math 411: Honours Complex Variables - University of Alberta

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

16 CHAPTER 3. POWER SERIES Figure 3.1: Surface plot of exp(z) in the complex plane, using an RGB color wheel to represent the phase. Red indicates real positive values. Theorem 3.1 (Radius of Convergence). Let � ∞ n=0 an(z − z0) n be a complex power series. Then there exists a unique R ∈ [0,∞] with the following properties: • � ∞ n=0 an(z −z0) n converges absolutely at each z ∈ BR(z0); • for each r ∈ [0,R), the series � ∞ n=0 an(z−z0) n converges uniformly on Br[z0] := {z ∈ C : |z −z0|≤ r}; • � ∞ n=0 an(z −z0) n diverges for each z /∈ BR[z0]. Moreover, R can be computed via the Cauchy–Hadamard formula: R = 1 � n limsup |an| n→∞ . It is called the radius of convergence for � ∞ n=0 an(z −z0) n .
Figure 3.2: Surface plot of cos(z) in the complex plane, using an RGB color wheel to represent the phase. Red indicates real positive values. Proof. The uniqueness of R follows from the first and the last property. Let R ∈ [0,∞] be defined by the Cauchy–Hadamard formula (we set 1 = ∞ and 0 1 = 0). ∞ Let r ∈ [0,R), and choose r ′ ∈ (r,R). It follows that � n limsup |an| = n→∞ 1 1 < R r ′, so that there exists n0 ∈ N such that n� |an| < 1 r ′ whenever n ≥ n0, i.e. � 1 |an|< r ′ �n for all n ≥ n0. For n ≥ n0 and z ∈ Br[z0], we then have |an(z −z0) n � r |≤ r ′ �n . 17
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: holds over C. We obtain: � ∞�
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 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
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• if w �= z: � � |g(w,z)−
Page 71 and 72:
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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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
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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
Page 103 and 104:
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
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Math 411: Honours Complex Variables - University of Alberta

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