Complex Analysis - Maths KU

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140 Chapter 2 <strong>Complex</strong> Functions and Mappings 33. If f(z) =z 1/6 is the principal 6th root function, then f(−1) = . 34. The complex reciprocal function 1/z is a composition of in the circle followed by reflection across the -axis. � � 2 35. According to the formal definition of a complex limit, lim z − i = −4 − i z→2i if for every ε > 0 there is a δ > 0 such that | | < ε whenever 0 < |z− | 0, −π
3 2 1 0 –1 –2 –3 –3 –2 –1 0 1 2 3 Level curves for f(z) = 1/z. See page 170. 3 Analytic Functions 3.1 Differentiability and Analyticity 3.2 Cauchy-Riemann Equations 3.3 Harmonic Functions 3.4 Applications Chapter 3 Review Quiz Introduction In the preceding chapter we introduced the notion of a complex function. Analogous to the calculus of real functions we can develop the notions of derivatives and integrals of complex functions based on the fundamental concept of a limit. In this chapter our principal focus will be on the definition and the properties of the derivative of a complex function. 141
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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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Page 111 and 112: 2.4 Special Power Functions 99 43.
Page 113 and 114: z c(z) Figure 2.41 Complex conjugat
Page 115 and 116: w = 1/z Figure 2.43 The reciprocal
Page 117 and 118: 2.5 Reciprocal Function 105 of the
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Page 125 and 126: 2.6 Limits and Continuity 113 Crite
Page 127 and 128: 2.6 Limits and Continuity 115 Befor
Page 129 and 130: 2.6 Limits and Continuity 117 Theor
Page 131 and 132: 2.6 Limits and Continuity 119 See (
Page 133 and 134: -1 y z = e iθ Figure 2.54 Figure f
Page 135 and 136: 2.6 Limits and Continuity 123 It th
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Page 139 and 140: y Values of arg decreasing Values o
Page 141 and 142: 2.6 Limits and Continuity 129 In Pr
Page 143 and 144: 2.6 Limits and Continuity 131 In Pr
Page 145 and 146: -4 -4 -3 -2 (-2, -1) 4 3 2 1 y -1 1
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Page 149 and 150: 4 2 y -4 -2 2 4 -2 -4 Figure 2.63 S
Page 151: Chapter 2 Review Quiz 139 10. The l
Page 155 and 156: 3.1 Differentiability and Analytici
Page 157 and 158: ☞ 3.1 Differentiability and Analy
Page 165 and 166: 3.2 Cauchy-Riemann Equations 153 We
Page 167 and 168: 3.2 Cauchy-Riemann Equations 155 EX
Page 169 and 170: 3.2 Cauchy-Riemann Equations 157 EX
Page 171 and 172: 3.3 Harmonic Functions 159 then sol
Page 173 and 174: 3.3 Harmonic Functions 161 EXAMPLE
Page 175 and 176: 3.3 Harmonic Functions 163 In Probl
Page 177 and 178: 3.4 Applications 165 tangent is the
Page 179 and 180: Lines of force ψ = c2 Lower potent
Page 181 and 182: In Section 4.5, for purposes that w
Page 183 and 184: y a b φ = k 1 x φ = k 0 Figure 3.
Page 185: Chapter 3 Review Quiz 173 11. The C
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190 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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428 Chapter 7 Conformal Mappings (a
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436 Chapter 7 Conformal Mappings φ
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438 Chapter 7 Conformal Mappings ψ
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444 Chapter 7 Conformal Mappings In
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448 Chapter 7 Conformal Mappings In
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Appendixes 1
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Appendix II Proof of the Cauchy-Gou
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Integrals � � � FE , GF , EG
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Appendix I Proof of Theorem 2.1 APP
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Appendix III Table of Conformal Map
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Answers to Selected Odd-Numbered Pr
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Indexes 1
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Word Index IND-7 Convergence of an
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Word Index IND-5 Cauchy-Riemann equ
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Symbol Index IND-3 z α , 194 sin z
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Word Index IND-9 principal square r
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IND-14 Word Index partial sums of,
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IND-12 Word Index Partial fractions
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Word Index IND-15 mapping by, 206-2
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Complex Analysis - Maths KU

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