Complex Analysis - Maths KU

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IND-10 Word Index evaluation by residues, 347, 352-353, 354-355, 357, 359 improper, 355, 359, 361-362 independent of the path, 265 line, 238-239 real, 236-240 Integral transform, 375 Integration along a branch cut, 361 Integration by parts, 270 Integration of power series, 314-315 Interior point of a set, 29 Invariance of Laplace’s equation under a mapping, 162, 223 Invariant set under a mapping, 78 Inverse Fourier transform: definition of, 381 evaluated by residue theory, 382-383 Inverse function, 88 Inverse hyperbolic functions, 219 derivatives of, 220 Inverse Laplace transform, 374 definition of, 377 evaluated by residue theory, 378 Inverse trigonometric functions, 215-216 derivatives of, 218 Inversion in the unit circle, 100 Irrotational flow, 167 Isolated singularity, 324 Isotherms, 167-168 Jordan’s inequality, 372 Joukowski airfoil, 442 Joukowski transformation, 442 Kernel of an integral transform, 375 Kinetic energy, 166 Lagrange’s identity, 23 Laplace transform: analyticity of, 377 definition of, 374 existence of, 376 inverse of, 374, 377-378 Laplace’s equation, 159, 222 invariance of, 162, 223 in polar coordinates, 163 Laplacian, 159 Laurent series, 326-327 analytic part of, 325 principal part of, 325 Laurent’s theorem, 327 Level curves, 164 L’Hôpital’s rule, 147, 149 proofs of, 151, 324 Limit(s): of a complex function, 112 imaginary part of, 116 infinite, 127 at infinity, 127 nonexistence of, 113 properties of, 117 of a real function, 111, 115 real part of, 116 of a sequence, 302-303 Linear approximation, 76 Linear fractional transformation: circle preserving property of, 402 composition of, 404 cross ratio for, 406 definition of, 400 on extended complex plane, 401 inverse of, 404-405 as a matrix, 404 Linear function, 68, 71 as a composition of a rotation, magnification, and translation, 72 as a magnification, 70 as a rotation, 69 as a translation, 68 Linear mappings, 68, 71-73 image of a point under, 72 Linear transformation, 384 Line equation of, 62 Line integrals in the plane, 238-241 Line(s): of flow, 167, 287 of force, 167-168 of heat flux, 167-168 parametrization of, 62 segment, 63 Liouville’s theorem, 279 Location of zeros, 366-367 Logarithmic function: algebraic properties of, 184 analyticity of, 187-188
IND-14 Word Index partial sums of, 303 power, 307 ratio test for, 306 remainder of, 317 root test for, 307 sum of, 303, 304 summation of, 367-368 Taylor, 315-316 Sets: annular region, 32 boundary of, 31 boundary points of, 31 bounded, 32 closed, 31 connected, 31 convex, 35 domain, 31 doubly connected, 257 exterior points of, 31 interior points of, 29 multiply-connected, 257 open, 29 polygonal, 410 region, 31 simply connected, 256 unbounded, 32 Simple: closed curve (contour), 237, 246 pole, 336 zero, 338 Simply connected domain, 256 Sine: hyperbolic, 209 inverse of, 214-215 Maclaurin series for, 318 as a mapping, 206-207 trigonometric, 200 zeros of, 205 Singularity, 146, 324 essential, 336 isolated, 318, 324 nonisolated, 325 removable, 336 pole, 336 Sink, 291, 294, 440-441 Skew-Hermitian matrix, 44 Smooth curve, 237, 246 Solenoidal vector field, 285 Source, 291, 294, 440-441 Speed, 135 Squre root function, 86 Stagnation point, 295 Steady-state: charge, 40 current, 40 temperature, 229 Stereographic projection, 33 Stream function, 167, 437 Streamlines, 135, 167-168, 287-288, 437-438 Streamlining, 437-438 Subtraction of complex numbers, 3-4 Sum of a series, 303 of a geometric series, 304 Summing infinite series by residues, 367-368 Table of conformal mappings, APP-9 Tangent function: hyperbolic, 209 trigonometric, 201 Taylor series, 315 Taylor’s formula with remainder, 317 Taylor’s theorem, 316 Temperatures, steady, 229, 231 Terminal point: of a curve, 237 of a vector, 10 Tests for convergence of series: ratio, 306 root, 307 Test point, 403 Transformations: conformal, 390-392 linear, 68, 71-73 linear fractional, 399-400 Schwarz-Christoffel, 412-413 Transform pairs, 375 Translation, 68 Triangle inequality, 12 Triangulated closed polygonal contour, APP-7 Trigonometric functions: analyticity of, 205-206 definitions of, 200-201 derivatives of, 206 identities for, 201-202 inverses of, 215-216
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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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444 Chapter 7 Conformal Mappings In
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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 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: Word Index IND-9 principal square r
Page 515 and 516: IND-12 Word Index Partial fractions
Page 517: Word Index IND-15 mapping by, 206-2
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