Math 411: Honours Complex Variables - University of Alberta

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

8 CHAPTER 1. THE COMPLEX NUMBERS and the trigonometric angle sum formulae. Notice that (cosθ,sinθ)·(cosφ,sinφ) = (cosθcosφ−sinθsinφ,cosθsinφ+sinθcosφ) = (cos(θ+φ),sin(θ+φ)). Thatis, multiplicationof2complexnumbersontheunitcirclex 2 +y 2 = 1corresponds to addition of their angles of inclination to the x axis. In particular, the mapping f(z) = z 2 doubles the angle of z = (x,y) and f(z) = z n multiplies the angle of z by n. These statements hold even if z lies on a circle of radius r �= 1: this is known as deMoivre’s Theorem. (rcosθ,rsinθ) n = r n (cosnθ,sinnθ);
Chapter 2 Complex Differentiation Definition. Let D ⊂ C, and let z0 be an interior point of D, i.e. there exists ǫ > 0 such that Bǫ(z0) := {z ∈ C : |z − z0|< ǫ} ⊂ D. A function f : D → C is called complex differentiable at z0 if exists. f ′ f(z)−f(z0) (z0) := lim z→z0 z −z0 Proposition 2.1. Let D ⊂ C, let z0 ∈ intD, and let f: D → C be complex differentiable at z0. Then f is continuous at z0. f(z)−f(z0) Proof. Since lim z→z0 z −z0 f(z)−f(z0) 0 = lim(z −z0) lim z→z0 z→z0 z −z0 so that f(z0) = lim f(z). z→z0 exists, we have = lim(z −z0) z→z0 f(z)−f(z0) z −z0 = lim(f(z)−f(z0)), z→z0 Proposition 2.2. Let D ⊂ C, and let f,g : D → C be complex differentiable at z0 ∈ intD. Then the following functions are complex differentiable at z0: f +g, fg, and, if g(z0) �= 0, f g and . Moreover, we have: (f +g) ′ (z0) = f ′ (z0)+g ′ (z0), (fg) ′ (z0) = f ′ (z0)g(z0)+f(z0)g ′ (z0), � � ′ f g (z0) = f′ (z0)g(z0)−f(z0)g ′ (z0) g(z0) 2 . 9
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: Proof. (i): If z = x+iy, then ¯z =
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 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
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
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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
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
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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))
Page 113 and 114:
Example. Let z be a complex number.
Page 115 and 116:
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
Page 121:
INDEX 121 straight line segment, 25
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Math 411: Honours Complex Variables - University of Alberta

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