Math 411: Honours Complex Variables - University of Alberta

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

Chapter 4 Complex Line Integrals We call a function f : [a,b] → C integrable if Ref,Imf : [a,b] → R are integrable in the sense of real variables. (The Riemann integral will do.) In this case, we define � b � b f(t)dt := a a � b Ref(t)dt+i Imf(t)dt. a Definition. A curve (or path) in C is a continuous map γ: [a,b] → C. We call • γ(a) the initial point of γ, • γ(b) the endpoint (or terminal point) of γ, and • {γ} := γ([a,b]) the trajectory of γ. Collectively, we call γ(a) and γ(b) the endpoints of γ. Examples. 1. Let z,w ∈ C. Then γ: [0,1] → C, t ↦→ z0 +t(z −z0) has the initial point z0 and the endpoint z, and {γ} is the line segment connecting z0 with z. 2. For k ∈ Z, let γk: [0,2π] → C, θ ↦→ e ikθ . Then γk(0) = 1 = γk(2π) holds, and for k �= 0, we have {γk} = {z ∈ C : |z|= 1}. Definition. A curve γ: [a,b] → C is called piecewise smoothif there exists a partition a = a0 < a1 < ··· < an = b such that γ|[aj−1,aj] is continuously differentiable for j = 1,...,n. 22
Definition. The length of a piecewise smooth curve γ: [a,b] → C is defined as ℓ(γ) := n� � aj |γ ′ (t)|dt, j=1 where a = a0 < a1 < ··· < an = b is a partition such that γ|[aj−1,aj] is continuously differentiable for j = 1,...,n. Definition. Let γ: [a,b] → C be a piecewise smooth curve, let a = a0 < a1 < ··· < an = b beapartitionsuch that γ|[aj−1,aj] iscontinuously differentiable forj = 1,...,n, and let f : {γ} → C be continuous. Then the line integral (or contour integral) of f along γ is defined as � γ � f := γ f(ζ)dζ = aj−1 n� � aj j=1 aj−1 f(γ(t))γ ′ (t)dt. Properties of the Line Integral. 1. Let γ be a piecewise smooth curve, let λ,µ ∈ C, and let f,g: {γ} → C be continuous. Then we have � � � (λf +µg) = λ f +µ g. γ 2. Let γ be a piecewise smooth curve, let f : {γ} → C be continuous, and let C ≥ 0 be such that |f(ζ)|≤ C for ζ ∈ {γ}. Then �� f� � ≤ Cℓ(γ) holds. γ 3. Let γ : [c,d] → C be a piecewise smooth curve, let φ : [a,b] → [c,d] be a continuously differentiable function with φ(a) = c and φ(b) = d, and let f : {γ} → C be continuous. Then we have � � f = f. γ 4. Let D ⊂ C be open, and let f : D → C be continuous with antiderivative F : D → C; i.e. F is complex differentiable at each z ∈ D, with F ′ (z) = f(z). Then � f = F(γ(b))−F(γ(a)) γ γ◦φ holds for every piecewise smooth curve γ: [a,b] → D. γ γ 23
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: Problem 3.1. Show in Theorem 3.2 th
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
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 and 78:
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
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
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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
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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))
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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
Page 119 and 120:
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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