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 3 Power Series Definition. A(complex)power seriesisaninfiniteseriesoftheform �∞ n n=0an(z−z0) withz,z0,a0,a1,a2,... ∈ C. Thepointz0 iscalledthepoint of expansionfortheseries. Examples. 1. For m ∈ N, we have m� n=0 z n = 1−zm+1 1−z if z �= 1. For |z|< 1, we obtain (letting m → ∞) 2. For z ∈ C, define Let z �= 0, and note that ∞� n=0 z n = 1 1−z . exp(z) := � � � z � n+1 �� z (n+1)! � n � � � n! � ∞� n=0 z n n! . = |z| n+1 → 0 as n → ∞. As the ratio test holds for series with summands in C as well as for series over R, we conclude that exp(z) converges absolutely. Let z,w ∈ C, and note that the Cauchy product formula for series over R also 14
holds over C. We obtain: � ∞� z exp(z)exp(w) = j=0 j �� ∞� j! k=0 ∞� n� z = n=0 k=0 n−k (n−k)! ∞� n� � 1 n = n! k n=0 k=0 n� (z +w) = n n! n=0 = exp(z +w). wk � k! w k k! � z n−k w k 15 by the Cauchy product formula, letting n = j +k, We call exp: C → C the exponential function. The above property suggests using the shorthand e z for exp(z). An interactive three-dimensional graph of exp(z) is shown in Figure 3.1. 3. The sine and cosine functions on C are defined as and sin(z) := cos(z) := ∞� n=0 (−1) n z 2n+1 (2n+1)! ∞� (−1) n=0 n z2n (2n)! for z ∈ C. As for exp(z), we see that both sin(z) and cos(z) converge absolutely for all z ∈ C. Moreover, we have for z ∈ C: e iz ∞� (iz) = n=0 n n! ∞� (iz) = n=0 2n (2n)! + ∞� (iz) n=0 2n+1 (2n+1)! ∞� n z2n = (−1) (2n)! +i ∞� (−1) n z2n+1 (2n+1)! n=0 = cos(z)+isin(z). Interactive three-dimensional graphs of the complex cosine and sine functions are shown in Figures 3.2, and 3.3. n=0
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: Example. Let Then f is totally diff
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
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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