Math 411: Honours Complex Variables - University of Alberta

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

18 CHAPTER 3. POWER SERIES Figure 3.3: Surface plot of sin(z) in the complex plane, using an RGB color wheel to represent the phase. Red indicates real positive values. Since r r ′ < 1, we have �∞ � ∞ n=0 � r r ′ �n < ∞. The Weierstraß M-test thus yields that n=0 an(z −z0) n converges absolutely and uniformly on Br[z0]. Since every z ∈ BR(z0) is contained in Br[z0] for some r ∈ [0,R), it follows that � ∞ n=0 an(z −z0) n converges absolutely for each such z. Let z /∈ BR[z0], i.e. |z −z0|> R, so that 1 1 < |z −z0| R and thus, for infinitely many n ∈ N, or, equivalently, = limsup n→∞ 1 |z −z0| < n� |an| 1 < |an(z −z0) n |. � n |an|
It follows that {an(z−z0) n } ∞ n=1 does not converge to zero. Consequently, � ∞ n=0 an(z− z0) n diverges. Examples. 1. � ∞ n=0 zn : R = 1. 2. � ∞ n=0 zn : R = ∞. n! 3. � ∞ n=0 n!zn : R = 0. 4. �∞ z2n+1 n=0 (−1)n (2n+1)! and �∞ z2n n=0 (−1)n : R = ∞. (2n)! Theorem 3.2 (Term-by-Term Differentiation). Let � ∞ n=0 an(z − z0) n be a complex power series with radius of convergence R. Then f: BR(z0) → C, z ↦→ ∞� an(z −z0) n n=0 is complex differentiable at each point z ∈ BR(z0) with f ′ (z) = ∞� nan(z −z0) n−1 . n=1 Proof. Without loss of generality, suppose that z0 = 0. We first show that � ∞ > limsup n→∞ 19 n=1nanz n−1 converges absolutely for each z ∈ BR(0). � n |an|, Let z ∈ BR(0), and choose r such that |z|< r < R. Since 1 r there exists n0 ∈ N such that |an|< � � 1 n for n ≥ n0 and thus r |nanz n−1 |< n � �n−1 |z| r r for n ≥ n0. Since |z| r < 1, we know from the Ratio Test that �∞ n n=1 r the Comparison Test then yields that � ∞ n=1 nanz n−1 converges absolutely. In view of the foregoing, we may define g: BR(0) → C, z ↦→ ∞� nanz n−1 . n=1 � �n−1 |z| < ∞; r We shall devote the rest of the proof to showing that f is complex differentiable on BR(0) with f ′ = g. To this end, fix ǫ > 0, and define, for z ∈ BR(0) and n ∈ N, Sn(z) := n� akz k k=0 and Rn(z) := ∞� k=n+1 akz k .
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: Figure 3.2: Surface plot of cos(z)
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,
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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:
Page 81 and 82:
12.2. THE GAMMA FUNCTION 81 12.2 Th
Page 83 and 84:
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
Page 97 and 98:
Proof. Of course, only (ii) =⇒ (i
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so that u(z) = � 2π u(re 0 iθ )
Page 101 and 102:
Set K := supζ∈∂D|f(ζ)|. For z
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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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