Math 411: Honours Complex Variables - University of Alberta

More documents

Recommendations

Info

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

116 CHAPTER 17. THE RIEMANN MAPPING THEOREM We also note that |h(z0)| 2 = |w| and define g: D → C, z ↦→ − |h′ (z0)| h ′ (z0) (φh(z0) ◦h)(z). Then g is injective with g(D) ⊂ D and g(z0) = 0. Since φ ′ a(a) = −1/(1−|a| 2 ), g ′ (z0) = − |h′ (z0)| h ′ (z0) ·φ′ h(z0)(h(z0))h ′ (z0) = |h′ (z0)| 1−|h(z0)| 2 = (1−|w|2 )f ′ (z0) 2 � |w|(1−|w|) = 1+|w| 2 � |w| ·f′ (z0) > f ′ (z0) > 0, so that g ∈ F and g ′ (z0) > f ′ (z0). This contradicts (∗). Theorem 17.4 (Simply Connected Domains). The following are equivalent for an open and connected set D ⊂ C: (i) D is simply connected; (ii) D admits holomorphic logarithms; (iii) D admits holomorphic roots; (iv) D admits holomorphic square roots; (v) D is all of C or biholomorphically equivalent to D; (vi) every holomorphic function f: D → C has an antiderivative; (vii) � f(ζ)dζ = 0 for each holomorphic function f : D → C and each closed curve γ γ in D; (viii) for every holomorphic function f: D → C, we have ν(γ,z)f(z) = 1 � 2πi γ for each closed curve γ in D and all z ∈ D \{γ}; f(ζ) ζ −z dζ (ix) every harmonic function u: D → R has a harmonic conjugate.
Proof. (i) =⇒ (ii) is Corollary 11.2.2, (ii) =⇒ (iii) is shown in the proof of Corollary 11.2.3, (iii) =⇒ (iv) is trivial, (iv) =⇒ (v) follows from Theorem 17.3, and (v) =⇒ (i) is implied by Proposition 17.1. (i) ⇐⇒ (vi) is Corollary 11.2.1 and (vi) ⇐⇒ (vii) follows from Theorem 4.1. (i) =⇒ (viii) follows from Theorem 11.1, and (viii) =⇒ (vii) is established in the proof of Theorem 11.2. (v) =⇒ (ix): Let u: D → R be harmonic. If D = C, the existence of a harmonic conjugate is immediate by Theorem 14.1. So suppose that D �= C. Hence, there is a biholomorphic map f: D → D. It is easily seen that ũ := u◦f −1 : D → R is harmonic and by Theorem 14.1 has a harmonic conjugate ˜v: D → R. Then v := ˜v ◦f: D → R is a harmonic conjugate of u. (ix) =⇒(ii): Letf: D → C beholomorphicsuch thatZ(f) = ∅. Thenu := log|f| is harmonic and thus has a harmonic conjugate v : D → R so that g := u + iv is holomorphic. On D we have so that |expg|= |exp(u+iv)|= expu = |f|, D → C, z ↦→ f(z) exp(g(z)) is a holomorphic function whose range lies on ∂D and therefore isn’t open. By the Open Mapping Theorem, this means that there exists c ∈ ∂D such that f(z) = cexp(g(z)) for z ∈ D. Choose θ ∈ R with exp(iθ) = c, and note that f(z) = exp(g(z)+iθ) for z ∈ D. Definition. Two (not necessarily piecewise smooth) closed curves γ1,γ2: [0,1] → D with γ1(0) = γ2(0) andγ1(1) = γ2(1) arecalled path homotopicifthere is acontinuous function Γ: [0,1]×[0,1] → D such that, for t ∈ [0,1] and for all s ∈ [0,1]. Γ(0,t) = γ1(t) and Γ(1,t) = γ2(t) Γ(s,0) = γ1(0) and Γ(s,1) = γ1(1) Definition. A closed curve γ is called homotopic to zero if γ and the constant curve γ(0) are path homotopic. Further Characterizations of Simply Connected Domains. There are further conditions that characterize simply connected domains. We will only state them, without giving proofs. Simple connectedness is also equivalent to: (x) every (not necessarily smooth) curve in D is homotopic to zero; (xi) D is homeomorphic to D. 117
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 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 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
Page 105 and 106: 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: By the Open Mapping Theorem, there
Page 119 and 120: Index C is a Field, 5 Biholomorphis
Page 121: INDEX 121 straight line segment, 25
show all

Math 411: Honours Complex Variables - University of Alberta

Create successful ePaper yourself

Delete template?

Save as template?