Math 411: Honours Complex Variables - University of Alberta

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

10 CHAPTER 2. COMPLEX DIFFERENTIATION Proof. As over R. Proposition 2.3. Let D,E ⊂ C, let g : D → C and f : E → C be such that g(D) ⊂ E, and let z0 ∈ intD be such that w0 := g(z0) ∈ intE. Further, suppose that g is complex differentiable at z0 and f is complex differentiable at w0. Then f◦g is complex differentiable at z0 with Proof. As over R. Examples. (f ◦g) ′ (z0) = f ′ (g(z0))g ′ (z0). 1. All constant functions are (on all of C) complex differentiable, as is z ↦→ z on C. Consequently, all complex polynomials are complex differentiable on all of C, and rational functions are complex differentiable wherever they are defined. 2. Let f: C → C, z ↦→ ¯z, and let z0 = x0+iy0 ∈ C. Assume that f is complex differentiable at z0. Then we have as well as f ′ ¯z − ¯z0 (z0) = lim z→z0 z −z0 (x0 −iy)−(x0 −iy0) = lim y→y0 (x0 +iy)−(x0 +iy0) i(y0 −y) = lim y→y0 i(y −y0) = −1 f ′ ¯z − ¯z0 (z0) = lim z→z0 z −z0 (x−iy0)−(x0 −iy0) = lim x→x0 (x+iy0)−(x0 +iy0) x−x0 = lim x→x0 x−x0 = 1, which is impossible. Hence, f is not complex differentiable at any z0 ∈ C. (On the other hand, f is continuously partially differentiable—as a function of two real variables—on all of C.) Lemma 2.1. The following are equivalent for an R-linear map T : C → C:
(i) there exists c ∈ C such that T(z) = cz for all z ∈ C; (ii) T is C-linear; (iii) T(i) = iT(1); (iv) the real 2×2 matrix representing T with respect to the standard basis of R2 may be written as � � a b A = −b a for some real a,b ∈ R. Proof. (i) =⇒ (ii) =⇒ (iii) is obvious. (iii) =⇒ (i): Set c := T(1). For z = x+iy ∈ C, this means that T(x+iy) = T(x)+T(iy) = xT(1)+yT(i) = xT(1)+iyT(1) = zT(1) = cz. (iv) ⇐⇒ (iii): Let a,b,c,d ∈ R be such that A = � a b c d represents T with respect to the standard basis of R2 . Note that � �� a b 1 a T(1) = = = a+ic, c d 0 c and Since we see that T(i) = � a b c d �� 0 1 � = � . � b d iT(1) = −c+ia, � = b+id. T(i) = iT(1) ⇐⇒ c = −b and d = a. Theorem 2.1 (Cauchy–Riemann Equations). Let D ⊂ C be open, and let z0 ∈ D. Let f: D → C and denote u := Ref, v := Imf. Then the following are equivalent: (i) f is complex differentiable at z0; 11
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: Chapter 2 Complex Differentiation D
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 +
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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
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• if w �= z: � � |g(w,z)−
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Corollary 11.2.1. Let D be an open,
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Examples. 1. Let f(z) = eiz z 2 +1
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12.1. APPLICATIONS OF THE RESIDUE T
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Page 79 and 80:
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12.2. THE GAMMA FUNCTION 81 12.2 Th
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12.2. THE GAMMA FUNCTION 83 Since
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12.2. THE GAMMA FUNCTION 85 6 5 4 3
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Chapter 13 Function Theoretic Conse
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Lemma 13.1. Let D ⊂ C be open and
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Definition. Let D ⊂ C be open, an
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Proof. Let p(z) = anz n +···+a1z
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for (x,y) ∈ R 2 \{(0,0)}. The par
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Proof. Of course, only (ii) =⇒ (i
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so that u(z) = � 2π u(re 0 iθ )
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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
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Chapter 16 Montel’s Theorem Defin
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Define g: D → RM as follows: for
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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
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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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