Math 411: Honours Complex Variables - University of Alberta

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

118 CHAPTER 17. THE RIEMANN MAPPING THEOREM Condition (xi) makes no reference to holomorphic functions and is entirely topological in nature. It means that there is a bijective, continuous map f: D → D with a continuous inverse. Since (x) is preserved under homeomorphisms, we see that (xi) implies (x). For the converse, it is sufficient to show that C is homeomorphic to D (for D �= C, this is clear by Theorem 17.3). Since and C → D, z ↦→ z 1+|z| D → C, z ↦→ z 1−|z| are continuous and inverse to each other, this is indeed the case. The converse to Problem 11.1 states that the property (xii) for every holomorphic function f: D → C, there exists a sequence of polynomials converging to f compactly on D always holds for a simply connected domain. The proof relies on Runge’s Approximation Theorem. There is also an equivalent condition for simple connectedness involving the extended complex plane C∞: (xiii) C∞ \D is connected.
Index C is a Field, 5 Biholomorphisms of D, 51 analytic, 28 analytic continuation, 105 angle preserving, 112 annulus, 57 Antiderivative Theorem, 27 Argument Principle, 90 Arzelà–Ascoli Theorem, 107 Bifurcation Theorem, 91 biholomorphic, 50 biholomorphically equivalent, 110 branch cut, 82 Casorati–Weierstraß Theorem, 55 Cauchy’s Integral Formula, 68 Cauchy’s Integral Formula for Circles, 35 Cauchy’s Integral Theorem, 70 Cauchy’s Integral Theorem for Annuli, 57 Cauchy’sIntegralTheoremforStar-Shaped Domains, 31 Cauchy–Hadamard formula, 16 Cauchy–Riemann Equations, 11 Cayley transform, 110 center, 30 closed, 24 Compact Convergence, 42 compact convergence, 43 complex conjugate, 6 complex differentiable, 9 complex numbers, 5 concatenation, 25 conformal, 50 ConformalityatNondegeneratePoints, 113 119 ConformalityofBiholomorphicMaps, 113 connected, 24 contour integral, 23 cosine, 15 curve, 22, 27 deMoivre’s Theorem, 8 discrete, 87 endpoint, 22 endpoints, 22 entire, 40 equicontinuous, 107 Equivalence ofHarmonicandMean-Value Properties, 102 essential, 55 exponential function, 15 extended complex numbers, 88 exterior, 66 Fractional Residue Theorem, 80 function element, 104 Fundamental Theorem of Algebra, 40, 92 Gamma Function, 81 Generalized Cauchy Integral Formula, 38 generalized Cauchy integral formula, 38 germ, 105 Goursat’s Lemma, 28 harmonic, 94 Harmonic Components, 94 harmonic conjugate, 95 Harmonic Conjugates, 95 Higher Derivatives, 38 Higher Derivatives of Power Series, 21 holomorphic, 28
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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
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: Proof. (i) =⇒ (ii) is Corollary 1
Page 121: INDEX 121 straight line segment, 25
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Math 411: Honours Complex Variables - University of Alberta

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