Math 411: Honours Complex Variables - University of Alberta

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

78 CHAPTER 12. THE RESIDUE THEOREM AND APPLICATIONS Problem 12.2. Let D ⊂ C be open, let f : D → C be holomorphic, and let z0 ∈ C\D be a simple pole of f. Show that res(gf;z0) = g(z0)res(f;z0) for every holomorphic function g: D ∪{z0} → C. Proposition 12.2 (Rational Functions). Let p and q be polynomials of one real variable with degq ≥ degp+2 and such that q(x) �= 0 for x ∈ R. Then we have � ∞ � p(x) p dx = 2πi� res q(x) q ,z � , where −∞ z∈H H := {z ∈ C : Imz > 0}. Proof. Since degq ≥ degp+2, the Comparison Test yields that the indefinite integral exists. For r > 0 consider the semicircle γr: [0,π] → C, θ ↦→ re iθ . Let ǫ > 0 be such that, for D := {z ∈ C : Imz > −ǫ}, we have Then D is simply connected and p q {z ∈ H : q(z) = 0} = {z ∈ D : q(z) = 0}. is holomorphic on D except at the zeros of q in H. For r large enough so that all zeros of q in D lie in the interior of [−r,r]⊕γr, we see by the Residue Theorem that � � p(ζ) p dζ = 2πi� res q(ζ) q ,z � = 2πi � � p res q ,z � . [−r,r]⊕γr z∈D Since degq ≥ degp+2, there exist numbers R > 0 and C ≥ 0 such that � � � � p(z) � � C �q(z) � ≤ |z| 2 γr z∈H for all z ∈ C with |z|≥ R. It follows that �� p(ζ) � q(ζ) dζ � � � C πC � ≤ πr sup ≤ ζ∈{γr} |ζ| 2 r for r ≥ R and thus � p(ζ) lim dζ = 0. r→∞ γr q(ζ)
12.1. APPLICATIONS OF THE RESIDUE THEOREM TO REAL INTEGRALS79 We then find that � ∞ Examples. 1. What is � ∞ 0 −∞ � p(x) dx = lim q(x) r→∞ 1 dx? 1+x 6 � = lim r→∞ � = lim r→∞ [−r,r] [−r,r] p(ζ) q(ζ) dζ p(ζ) dζ + lim q(ζ) r→∞ p(ζ) [−r,r]⊕γr q(ζ) dζ = 2πi � � p res q ,z � . z∈H � γr p(ζ) q(ζ) dζ The zeros of q(z) = 1 + z6 are of the form eiθ where θ ∈ [0,2π) is such that ei6θ = −1 = eiπ , i.e. 6θ − π ∈ 2πZ, so that θ = π π 5π 7π 3π 11π , , , , , . For 6 2 6 6 2 6 k = 1,...,6, let zk = e i(2k−1)π 6. Then 1 q has a simple pole at zk for k = 1,...,6. By Problem 12.1, we have res so that by, Proposition 12.2, � ∞ 0 1 1 dx = 1+x 6 2 � 1 q ,zk � � ∞ = 1 q ′ 1 = (zk) 6z5 k 1 dx −∞ 1+x 6 3� � 1 = πi res q k=1 ,zk � = − πi � 6 = − πi � cos 6 π π +isin 6 6 = π � 2sin 6 π 6 +1 � = π 3 . e iπ 6 +e iπ 2 +e i5π 6 � = − zk 6 , +i+cos 5π 6 � 5π +isin 6
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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
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
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 77: 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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