Complex Analysis - Maths KU

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IND-8 Word Index Entire function, 145 Epsilon-delta proofs, 113-114 ε−neighborhood, 112, 302 Equality of complex numbers, 2-3 Equipotential curves, 167-168 Error function, 322 Essential singularity, 336 Euclidean isometry, 79 Euler’s formula, 39 Evaluation of real integrals by residues, 352, 354, 357, 359 Even function, 355 Exact first-order differential equation, 288 Exponential form of a complex number, 39-40 Exponential function: algebraic properties of, 56, 178 analyticity of, 177 definition of, 39, 55, 176 derivative of, 177 fundamental region for, 180 inverse of, 186-187 Maclaurin series for, 318 mapping properties of, 180-181 modulus of, 178 periodicity of, 54, 179 Exponential order, 376 Extended: complex number system, 33 complex plane, 103 real number system, 32 Exterior point of a set, 31 Factorization of a quadratic polynomial, 38 Fixed point, 78 Flow around a unit circle, 138 Fluid flow: circulation of, 290-292 ideal, 285, 287 incompressible, 167, 285 irrotational, 167, 284 net flux of, 290-292 planar, 135, 284 sink of, 291, 294 source of, 291, 294 stagnation point of, 295 streamlines for, 135, 286, 287 velocity field for, 135, 284 Flux, net, 290, 292 Flux lines, 167 Fourier integrals, 357 Fourier transform, 381 Function(s): analytic, 145 bounded, 124 branch of, 187, 197, 217 branch point of, 127, 188 complex, 50 complex conjugation, 101 composition of, 71-72, 78 constant, 117, 156 continuous, 120, 122 definition of, 50 derivative of, 142 differentiable, 142 discontinuous, 120 domain of, 50 entire, 145 exponential, 39, 53, 55, 176 of exponential order, 276 harmonic, 160 harmonic conjugate, 160 holomorphic, 145 hyperbolic, 209 identity, 117 image under, 50 imaginary part of, 51-52 input of, 50 integrable, 237, 248 inverse of, 88 inverse hyperbolic, 219 inverse trigonometric, 215-216 limit of, 120 linear, 68, 71 logarithmic, 182, 185 meromorphic, 340 multiple-valued, 94-95 multiple-to-one, 95 one-to-one, 88 output of, 50 periodic, 179 polar coordinate description of, 54 polynomial, 80, 146 power, 194-196 principal branch of, 187, 197 principal nth root, 93
Word Index IND-7 Convergence of an improper integral, 356 Convergence of an infinite series: absolute, 306 conditional, 306 definition of, 303 necessary condition for, 305 tests for, 306-307 Convergence of a power series, 307 circle of, 307 radius of, 307-308, 318 tests for, 306-307 Convergence of a sequence, 302-303 in terms of real and imaginary parts, 303 Convex set, 35 Cosine: hyperbolic, 209 inverse of, 216 Maclaurin series for, 318 as a mapping, 208 trigonometric, 200, 204 zeros of, 205 Critical point, 393 Crosscut, 259, 261 Cross ratio, 406 invariance of, 406-407 Commutative laws, 3 Cubic formula, 44 Curl of a vector field, 167, 285 Curve: closed, 237, 246 initial point of, 237 opposite, 246 oriented, 246 piecewise smooth, 237, 246 simple closed, 237, 246 smooth, 237, 246 terminal point of, 237 Definite integral, 236-237 Deformation of contours, 259 Deleted neighborhood, 29 de Moivre’s formula, 20 Depressed cubic equation, 44 Derivative: of complex exponential function, 177 of complex hyperbolic functions, 209-210 of complex inverse hyperbolic functions, 220 of complex inverse trigonometric functions, 218 of the complex logarithm, 188 of complex trigonometric functions, 206 definition of, 142, 146 evaluation of, 142 formula for, 155 of power series, 314 rules for, 143 symbols for, 142 Differentiability implies continuity, 146 Differentiable at a point, 142 Differential equation, 38 Differentiation of a power series, 314 Differentiation, rules of, 143 Dirichlet problem, 168, 420, 429 in a half-plane, 228, 420-421 steps for solving, 225, 429 for unit disk, 425-426 Discontinuous at a point, 120 Disk, 29 closed, 31 open, 31 punctured, 32 Distance between two points, 11 Distributive law, 3 Divergence: of improper integrals, 354 of sequences, 302 of an infinite series, 304, 305, 306, 307 of a vector field, 167, 285 Division of complex numbers, 3, 5 in polar coordinates, 18 Domain: connected, 31 definitions of, 31, 50 doubly connected, 257 of a function, 50 multiply connected, 257 simply connected, 256 triply connected, 257 Electrostatic potential, 166, 432
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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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432 Chapter 7 Conformal Mappings 3
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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 454 and 455: 442 Chapter 7 Conformal Mappings 4
Page 456 and 457: 444 Chapter 7 Conformal Mappings In
Page 458 and 459: 446 Chapter 7 Conformal Mappings 24
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 503: Indexes 1
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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