Complex Analysis - Maths KU

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ANS-10 Answers to Selected Odd-Numbered Problems Exercises 2.5, page 107 1. |w| = 1 5 3. |w| = 1 , −3π/4 ≤ arg(w) ≤ π/4 3 5. 1 1 ≤|w| ≤3 7. arg(w) =− 2 4 π 9. � �w + 1 8 i�� = 1 8 13. u = 1 4 11. v = 1 2 15. Image is the region bounded by � �w + 1 � � = 4 1 4 and � �w + 1 � � = 2 1 2 . 17. Image is the region bounded by v = 0, v = −u, and |w| = 2 and containing the point −3+2i 19. (a) Invert in the unit circle, reflect across the real axis, rotate through π/2 counterclockwise about the origin, magnify by 2, then translate by 1. (b) � �w − 1 − 1 4 i�� = 1 4 (c) v = − 1 2 21. (a) if f(z) =1/z and g(z) =z 2 , then h(z) =g(f(z)) (b) u = 1 4 v2 − 1 (c) u = 1 − v2 4 Exercises 2.6, page 128 1. −4+2i 3. 3 − i 5. −1 7. 2+e 2 cos 1 + i � 1+e 2 sin 1 � ≈ 5.9923 + 7.2177i 9. 1 − 3i 11. √ 2 13. 4i 15. a 17. (a) 1 (b) 0 (c) does not exist 19. (a) 1 (b) 1 (c) no 21. 1 5 (d) −1 (e) does not exist 2 − i 23. ∞ 5 25. ∞ 27. lim f(z) =f(2 − i) =5− 8i z→2−i 29. lim f(z) =f(i) = z→i 1 3 i 31. limf(z) =f(1)=3 z→1 33. lim f(z) =f(3 − 2i) =−6+3i 35. f(−i) is not defined z→3−2i 37. lim f(z) does not exist 39. limf(z) does not exist z→−1 41. the entire complex plane C 43. all points in the complex plane except those on the circle |z| =2 z→i
Answers to Selected Odd-Numbered Problems ANS-23 7. φ(x, y) = y π + x2 − y 2 � tan π −1 � � x − 1 − tan y −1 � �� x y + xy π log � 2 2 (x − 1) + y e x2 + y2 � 9. (b) φ(x, y) =e −y cos x Exercises 7.5, page 442 1. (a) w = z 2 (b) φ(x, y) = 1 � � � � 2 2 −Arg z +1 − Arg z π � + 2Arg � z 2 − 1 �� 2 1+z 3. (a) w = 1 − z (b) φ(x, y) =1+ 1 � �� 2 � � � 2 1+z 1+z 2Arg +1 +Arg π 1 − z 1 − z �� 2 1+z −2Arg − 1 1 − z � π 5. (a) w = sin 4 z � (b) φ(x, y) =1+ 1 � � � π −3Arg sin π 4 z � � � � π +1 +3Arg sin 4 z �� π −Arg sin 4 z � �� − 1 7. (a) w = 1 (b) φ(x, y) = z −2x x2 +2 + y2 2z − 1 − 9. (a) w = √ 3 � √ � √ 4+2 3 z +1+ 3 � � 10 � 2z − 1 − (b) φ(x, y) = � √ � loge � loge 7 − 4 3 � √ � � 3 � � √ � √ � 4+2 3 z +1+ 3 � 11. (a) w = sin −1 z (b) φ(x, y) =5+ 10 π Re� sin −1 z � 13. Ω(z) =z 4 17. (a) z = πi − 1 [Ln(w + 1)+ Ln(w − 1)] 2 (b) z(t) =πi − 1 2 15. Ω(z) = cosh z [Ln(t +1+ic2)+ Ln(t − 1+ic2)] 19. (a) z = 1 ��w2 � � 1/2 −1 − 1 + cosh w π (b) z(t) = 1 ��(t 2 � � 1/2 −1 + ic2) − 1 + cosh (t + ic2) π 21. Ω(z) =Ln � z 4 +4 � − Ln � z 4 − 16 � Chapter 7 Review Quiz, page 448 1. false 3. false 5. true 7. true 9. false 11. true 13. true 15. true
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A First Course in Complex Analysis
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For Dana, Kasey, and Cody
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vi Contents Chapter 4. Elementary F
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Preface 7.2 Preface Philosophy This
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Preface xi to, but not formally cov
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3π 2π π 0 -π -2π -3π -1 0 1 -
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1.1 Complex Numbers and Their Prope
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y z 2 z 2 - z 1 or (x 2 - x 1 , y 2
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1.2 Complex Plane 13 From (8) with
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1.2 Complex Plane 15 30. Find an up
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O y θ x = r cos θ (r, θ) or (x,
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1.3 Polar Form of Complex Numbers 1
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1.3 Polar Form of Complex Numbers 2
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1.4 Powers and Roots 23 arg(z) =π/
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√ 4 = 2 and 3 √ 27 = 3 are the
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1.4 Powers and Roots 27 16. Rework
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z 0 ρ ρ |z - z 0 |= Figure 1.15 C
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y Interior Exterior Boundary Figure
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1.5 Sets of Points in the Complex P
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1.5 Sets of Points in the Complex P
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Note: The roots z1 and z2 are conju
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1.6 Applications 39 c = 0.The latte
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1.6 Applications 41 Electrical engi
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1.6 Applications 43 In Problems 25
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Chapter 1 Review Quiz 45 Here the s
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Chapter 1 Review Quiz 47 27. The pr
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3i 2i i S 1 2 3 Image of a square u
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2.1 Complex Functions 51 interchang
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2.1 Complex Functions 53 Definition
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2.1 Complex Functions 55 respective
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2.1 Complex Functions 57 Focus on C
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2.2 Complex Functions as Mappings 5
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-4 -4 -3 C 2 3 4 (a) The vertical l
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4 2 -6 -4 -2 2 4 6 -2 -4 v Figure 2
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3i 2i i S z 1 2 3 S′ T(z) Figure
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ar Figure 2.12 Magnification C C′
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Note: The order in which you perfor
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2.3 Linear Mappings 75 triangle S4
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2.3 Linear Mappings 77 In Problems
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2.3 Linear Mappings 79 36. (a) Give
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2θ r θ r 2 Figure 2.17 The mappin
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-3 y 3 2 1 -2 -1 1 2 3 -1 -2 -3 (a)
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y 2 1.5 1 0.5 -2 -1.5 -1 -0.5 -0.5
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Note ☞ 2.4 Special Power Function
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Solve the equation z = f(w) for w t
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Remember: Arg(z) is in the interval
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S y (a) A circular sector 3π/8 v 3
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B y y A w = z 2 x (a) A maps onto A
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3π 2π π 0 -π -2π -3π -1 0 1 -
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2.4 Special Power Functions 99 43.
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z c(z) Figure 2.41 Complex conjugat
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w = 1/z Figure 2.43 The reciprocal
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2.5 Reciprocal Function 105 of the
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2.5 Reciprocal Function 107 underst
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2.5 Reciprocal Function 109 24. Acc
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L y y = L + ε y = L - ε y = f(x)
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2.6 Limits and Continuity 113 Crite
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2.6 Limits and Continuity 115 Befor
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2.6 Limits and Continuity 117 Theor
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2.6 Limits and Continuity 119 See (
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-1 y z = e iθ Figure 2.54 Figure f
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2.6 Limits and Continuity 123 It th
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2.6 Limits and Continuity 125 While
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y Values of arg decreasing Values o
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2.6 Limits and Continuity 129 In Pr
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2.6 Limits and Continuity 131 In Pr
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-4 -4 -3 -2 (-2, -1) 4 3 2 1 y -1 1
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Note: Normalized vector fields shou
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4 2 y -4 -2 2 4 -2 -4 Figure 2.63 S
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Chapter 2 Review Quiz 139 10. The l
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3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3 L
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3.1 Differentiability and Analytici
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☞ 3.1 Differentiability and Analy
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3.2 Cauchy-Riemann Equations 153 We
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3.2 Cauchy-Riemann Equations 155 EX
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3.2 Cauchy-Riemann Equations 157 EX
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3.3 Harmonic Functions 159 then sol
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3.3 Harmonic Functions 161 EXAMPLE
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3.3 Harmonic Functions 163 In Probl
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3.4 Applications 165 tangent is the
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Lines of force ψ = c2 Lower potent
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In Section 4.5, for purposes that w
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y a b φ = k 1 x φ = k 0 Figure 3.
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Chapter 3 Review Quiz 173 11. The C
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176 Chapter 4 Elementary Functions
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Normalized velocity vector field fo
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5.1 Real Integrals 237 3. Let �P
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y π t = gives 2 (0,4) C t = 0 give
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(-1, -1) y C (2, 8) x Figure 5.5 Gr
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5.1 Real Integrals 243 denotes the
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5.2 Complex Integrals 245 In Proble
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z 0 C z k * z 1 * z 2 * z 2 z 1 z k
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These properties are important in t
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5.2 Complex Integrals 251 Now in vi
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5.2 Complex Integrals 253 Proof It
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y 1 + i 1 Figure 5.21 Figure for Pr
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D C Figure 5.26 Simply connected do
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D D A C C (a) (b) B C 1 C 1 Figure
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C 1 C 1 C (a) C (b) Figure 5.31 Tri
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C C y Figure 5.34 Figure for Proble
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z 0 C 1 D z 1 C Figure 5.38 If f is
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5.4 Independence of Path 267 and z(
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Be careful when using Ln z as an an
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5.4 Independence of Path 271 (ii) I
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5.5 Cauchy’s Integral Formulas an
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3i -3i y Figure 5.44 Contour for Ex
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y i 0 C 2 C 1 Figure 5.45 Contour f
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5.6 Applications 285 A B A B A B (a
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5.6 Applications 287 Indeed, by rew
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Note ☞ 5.6 Applications 289 If yo
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-2 -1 2 1 -1 -2 y Figure 5.51 Zero
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-1 -0.5 1 0.5 -0.5 -1 y 0.5 Figure
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5.6 Applications 295 In Problems 5-
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C 1 y 2i C 2 -2 2 Figure 5.58 Figur
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Chapter 5 Review Quiz 299 � z 38.
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302 Chapter 6 Series and Residues y
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304 Chapter 6 Series and Residues Y
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306 Chapter 6 Series and Residues D
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308 Chapter 6 Series and Residues E
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310 Chapter 6 Series and Residues R
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312 Chapter 6 Series and Residues 3
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314 Chapter 6 Series and Residues I
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316 Chapter 6 Series and Residues D
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318 Chapter 6 Series and Residues N
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326 Chapter 6 Series and Residues a
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328 Chapter 6 Series and Residues N
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330 Chapter 6 Series and Residues T
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334 Chapter 6 Series and Residues R
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336 Chapter 6 Series and Residues i
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338 Chapter 6 Series and Residues A
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340 Chapter 6 Series and Residues S
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344 Chapter 6 Series and Residues T
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346 Chapter 6 Series and Residues (
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354 Chapter 6 Series and Residues H
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356 Chapter 6 Series and Residues -
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358 Chapter 6 Series and Residues o
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360 Chapter 6 Series and Residues -
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362 Chapter 6 Series and Residues z
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364 Chapter 6 Series and Residues f
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366 Chapter 6 Series and Residues v
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368 Chapter 6 Series and Residues
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378 Chapter 6 Series and Residues C
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380 Chapter 6 Series and Residues s
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386 Chapter 6 Series and Residues P
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390 Chapter 7 Conformal Mappings y
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394 Chapter 7 Conformal Mappings π
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396 Chapter 7 Conformal Mappings Re
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398 Chapter 7 Conformal Mappings 15
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400 Chapter 7 Conformal Mappings De
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402 Chapter 7 Conformal Mappings Th
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404 Chapter 7 Conformal Mappings v
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406 Chapter 7 Conformal Mappings Re
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408 Chapter 7 Conformal Mappings No
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412 Chapter 7 Conformal Mappings x
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414 Chapter 7 Conformal Mappings A
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416 Chapter 7 Conformal Mappings fo
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418 Chapter 7 Conformal Mappings EX
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420 Chapter 7 Conformal Mappings
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Page 438 and 439: 426 Chapter 7 Conformal Mappings y
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Page 448 and 449: 436 Chapter 7 Conformal Mappings φ
Page 450 and 451: 438 Chapter 7 Conformal Mappings ψ
Page 456 and 457: 444 Chapter 7 Conformal Mappings In
Page 460 and 461: 448 Chapter 7 Conformal Mappings In
Page 463 and 464: Appendixes 1
Page 465 and 466: Appendix II Proof of the Cauchy-Gou
Page 467 and 468: Integrals � � � FE , GF , EG
Page 469 and 470: Appendix I Proof of Theorem 2.1 APP
Page 471 and 472: Appendix III Table of Conformal Map
Page 479 and 480: Answers to Selected Odd-Numbered Pr
Page 487: Answers to Selected Odd-Numbered Pr
Page 503 and 504: Indexes 1
Page 505 and 506: Word Index IND-7 Convergence of an
Page 507 and 508: Word Index IND-5 Cauchy-Riemann equ
Page 509 and 510: Symbol Index IND-3 z α , 194 sin z
Page 511 and 512: Word Index IND-9 principal square r
Page 513 and 514: IND-14 Word Index partial sums of,
Page 515 and 516: IND-12 Word Index Partial fractions
Page 517: Word Index IND-15 mapping by, 206-2
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Complex Analysis - Maths KU

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