Math 411: Honours Complex Variables - University of Alberta

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

120 INDEX holomorphic nth roots, 71 Holomorphic Inverses, 113 Holomorphic Logarithms, 71 holomorphic logarithms, 71 Holomorphic Roots, 71 holomorphic roots, 71 homologous to zero, 68 homotopic to zero, 117 Hurwitz’s Theorem, 91 Identity Theorem, 46 imaginary part, 6 imaginary unit, 6 initial point, 22 integrable, 22 Integration of Power Series, 21 interior, 66 isolated singularity, 52 Jordan’s Lemma:, 80 Laurent Coefficients, 61 Laurent Decomposition, 59 Laurent decomposition, 61 Laurent series, 61 length, 23 line integral, 23 Liouville’s Theorem, 40 Local Uniform Convergence, 42 locally constant, 24 Locally Constant vs. Connectivity, 25 locally uniformly, 42 Maximum Modulus Principle, 48 MaximumModulusPrincipleforBounded Domains, 49 Mean Value Equation, 36 mean value property, 101 meromorphic function, 88 Meromorphic Functions Form a Field, 89 Monodromy Theorem, 105 Montel’s Theorem, 109 Morera condition, 39 Open Mapping Theorem, 48 order, 54 ordered, 6 orthogonal, 112 path, 22 path homotopic, 117 piecewise smooth, 22 point of expansion, 14 Poisson kernel, 98 Poisson’s Integral Formula, 98 pole, 53 power series, 14 Power Series for Holomorphic Functions, 45 principal argument, 31 principal branch of the logarithm, 31 principal part, 61 Radius of Convergence, 16 radius of convergence, 16 Rational Functions, 78 Rational Trigonometric Polynomials, 75 real part, 6 region, 26 removable, 52 residue, 72 Residue Theorem, 73 reversed curve, 25 Riemann Mapping Theorem, 114 Riemann sphere, 88 Riemann’s Removability Condition, 52 Rouché’s Theorem, 92 Runge’s Approximation Theorem, 118 Schwarz’s Lemma, 49 secondary part, 61 simple pole, 54 simply connected, 68 Simply Connected Domains, 116 sine, 15 sliced plane, 31 star shaped, 30
INDEX 121 straight line segment, 25 Term-by-Term Differentiation, 19 terminal point, 22 trajectory, 22 Uniform Convergence Preserves Continuity, 42 uniformly, 42 Weierstraß Theorem, 44 winding number, 65 Winding Numbers Are Locally Constant, 67 zero, 56
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Math 411: Honours Complex Variables
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Contents 1 The Complex Numbers 5 2
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Chapter 1 The Complex Numbers Defin
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Proof. (i): If z = x+iy, then ¯z =
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Chapter 2 Complex Differentiation D
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(i) there exists c ∈ C such that
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Example. Let Then f is totally diff
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holds over C. We obtain: � ∞�
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Figure 3.2: Surface plot of cos(z)
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It follows that {an(z−z0) n } ∞
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Problem 3.1. Show in Theorem 3.2 th
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Definition. The length of a piecewi
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1. Given a < b < c and two curves
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That is, F ′ (z) = f(z). Theorem
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∆ (4) ∆ (1) ∆ (2) As the line
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Proof. Let z0 ∈ D be a center for
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four subtriangles, denoted by ∆(z
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γ1 Then it is clear that � � 1
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Proof. There is no loss of generali
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Proof. Apply Theorem 5.3 and 5.4. E
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is entire. Since p is a nonconstant
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(ii) for each compact K ⊂ D, the
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Proof. Let ζ ∈ ∂Br(z0), and no
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holds for some a0,a1,a2,... ∈ C a
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Proof. Let z0 ∈ D be such that |f
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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
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: Index C is a Field, 5 Biholomorphis
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Math 411: Honours Complex Variables - University of Alberta

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