Complex Analysis - Maths KU

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IND-6 Word Index at infinity, 32 integer powers of, 19 modulus of, 10 multiplication of, 3-4 nth power of, 19-20 polar form of, 16 principle argument of, 17 principle nth root of, 25 pure imaginary, 2 rational powers of, 26-27 real part of, 2 reciprocal of, 6 roots of, 24 system C, 3 subtraction of, 3-4 triangle inequality for, 12 vector interpretation, 10-11 Complex parametric curve, 62, 246 Complex plane, 10 distance in, 11 imaginary axis of, 10 real axis of, 10 Complex potential function, 167, 227 Complex power function, 80 analyticity, 197 definition of, 195 derivative of, 197 Complex powers, 194 principal value of, 196 Complex representation of a vector field, 133 Complex sequence, 302 convergence of, 302-303 divergence of, 302 Complex series, 303 Complex squaring function, 81 Complex trigonometric functions, 200-201 Complex-valued function of a complex variable, 50 Complex-valued function of a real variable, 56, 248 Complex velocity, 289 Complex velocity potential, 167 Composition of functions, 71-72, 78 transformations, 404 Conformal mapping(s): definition of, 391 and the Dirichlet problem, 429 and the Neumann problem, 434-435 streamlining, 437-438 table of, APP-9 Conjugate of a complex number, 4-5 Connected set, 31 Conservation of energy, 166 Conservative vector fields, 166 Constant map, 71 Continuity: of a complex function, 120 at a point, 120 of polynomial functions, 123 of rational functions, 124 of a real function, 119, 122 on a set, 122 Continuous functions: bounding property of, 124 imaginary part of, 122 properties of, 123 real part of, 122 Continuous parametric curve, 128 Contour: closed, 237, 246 definition of, 246 deformation of, 259 indented, 359 initial point of, 237, 246 length of, 252 negative direction of, 246 opposite, 246 orientation of, 246 parametrization of, 246 piecewise smooth, 237, 246 positive direction on, 246 simple, 237, 246 simple closed, 237, 246 smooth, 237, 246 terminal point of, 237, 246 Contour integral: behavior of, 357, 359 bounding theorem for, 252 definition of, 247-248 evaluation of, 249-250 fundamental theorem for, 266 properties of, 251-252 Contraction, 71 Contrapositive of a proposition, 154
Word Index IND-5 Cauchy-Riemann equations: derivation of, 152-153 necessity of, 152 in polar form, 156 sufficiency of, 154-155 Cauchy’s inequality, 278 Cauchy’s integral formulas: for derivatives, 275 for a function, 273 Center of mass of a wire, 245 Center of a power series, 307 Chain rule of differentiation, 143 Circle: equation of, 29 parametrization of, 63 Circle of convergence, 307 Circle-preserving property, 402 Circulation of velocity field, 290-292 Closed: curve (contour), 237, 248 disk, 31 region, 31 Complex conjugate, 4 Complex conjugation function, 101 Complex constant function, 117 Complex exponential function: algebraic properties of, 54, 178 analyticity of, 177 definition of, 39, 53, 176 derivative of, 177 mapping properties of, 181 periodicity of, 54, 179 Complex function: analytic, 145 bounded, 124 branches of, 125 conjugation, 101 constant, 117, 156 continuous, 120 definition of, 50 derivative of, 142, 146, 155 differentiable, 142 domain of, 50 entire, 145 exponential, 39, 53, 176 hyperbolic, 209 identity, 117 imaginary part of, 52 inverse hyperbolic, 219 inverse trigonometric, 215-216 limit of, 112 linear, 68, 71 logarithmic, 182 as a mapping, 58 polar coordinate form of, 54 polynomial, 80, 146 power, 194 principal square root, 86 real part of, 52 range of, 50 rational, 100, 146 reciprocal, 100 squaring, 81 square root, 86 trigonometric, 200-201 as a two-dimensional fluid flow, 133 velocity, 289 Complex impedance, 41 Complex integral: definition of, 247 evaluation of, 249-250 Complex logarithm: algebraic properties of, 184 analyticity of, 187-188 definition of, 182 derivative of, 188 branch cut for, 187 principal branch of, 187 principal value of, 184-185 Complex mapping, 58 Complex matrices, 44 Complex number(s): absolute value of, 10 addition of, 3-4 argument of, 17 associative laws for, 3 commutative laws for, 3 complex powers of, 194 conjugate of, 4-5 definition of, 2 distance between, 11 distributive law for, 3 division of, 3, 5 equality of, 2-3 exponential form of, 39-40, 53 geometric interpretations of, 10 imaginary part of, 2
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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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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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Page 456 and 457: 444 Chapter 7 Conformal Mappings In
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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 503 and 504: Indexes 1
Page 505: Word Index IND-7 Convergence of an
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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