Complex Analysis - Maths KU

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412 Chapter 7 Conformal Mappings x 1 y x 2 (a) The upper half-plane y ≥ 0 π – α 1 v α 1 α 2 w = f(z) π – α 2 (b) The image of the region in (a) u x An argument of w ′ (t) is then given by: � � � � α1 α2 Arg (A)+ − 1 Arg (t − x1)+ − 1 Arg (t − x2) . (5) π π Because −∞
A i –i v y –1 1 (a) Half-plane y ≥ 0 π/2 π/2 B′ A′ (b) Semi-infinite strip Figure 7.23 Figure for Example 1 B x u Note ☞ 7.3 Schwarz-Christoffel Transformations 413 It follows from Theorem 7.1 of Section 7.1 that the function given bythe Schwarz-Christoffel formula (6) is a conformal mapping in the domain y>0. For the sake of brevity, we will, henceforth, refer to a mapping obtained from (6) as a conformal mapping from the upper half-plane onto a polygonal region. It should be kept in mind that although such a mapping is defined on the upper half-plane y ≥ 0, it is onlyconformal in the domain y>0. Before investigating some examples of the Schwarz-Christoffel formula, we need to point out three things. First, in practice we usuallyhave some freedom in the selection of the points xk on the x-axis. A judicious choice can simplifythe computation of f(z). Second, Theorem 7.4 provides a formula onlyfor the derivative of f. A general formula for f is given byan integral � f(z) =A (z − x1) (α1/π)−1 (z − x2) (α2/π)−1 ···(z − xn) (αn/π)−1 dz + B, where A and B are complex constants. Thus, f is the composition of the function � g(z) = (z − x1) (α1/π)−1 (z − x2) (α2/π)−1 ···(z − xn) (αn/π)−1 dz and the linear mapping h(z) =Az +B. As described in Section 2.3, the linear mapping h allows us to rotate, magnify(or contract), and translate the polygonal region produced by g. Third, although it is not stated in Theorem 7.4, the Schwarz-Christoffel formula (6) can also be used to construct a mapping of the upper half-plane y ≥ 0 onto a bounded polygonal region. To do so, we apply(6) using onlyn − 1ofthen interior angles of the bounded polygonal region. † We illustrate these ideas in the following examples. EXAMPLE 1 Using the Schwarz-Christoffel Formula Use the Schwarz-Christoffel formula (6) to construct a conformal mapping from the upper half-plane onto the polygonal region defined by u ≥ 0, −1 ≤ v ≤ 1. Solution Observe that the polygonal region defined by u ≥ 0, −1 ≤ v ≤ 1, is the semi-infinite strip shown in grayin Figure 7.23(b). The interior angles of this unbounded polygonal region are α1 = α2 = π/2, and the vertices are w1 = −i and w2 = i. To find the desired mapping, we applyTheorem 7.4 with x1 = −1 and x2 = 1. With these identifications, (6) gives f ′ (z) =A (z +1) −1/2 (z − 1) −1/2 . (7) From Theorem 7.4, w = f(z) is a conformal mapping from the half-plane y ≥ 0 onto the polygonal region u ≥ 0, −1 ≤ v ≤ 1. A formula for f(z) is found byintegrating (7). Since z is in the upper half-plane y ≥ 0, we first use † For a bounded polygon in the plane, any n − 1 of its interior angles uniquely determine the remaining one.
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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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324 Chapter 6 Series and Residues I
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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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348 Chapter 6 Series and Residues E
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352 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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Page 398 and 399: 386 Chapter 6 Series and Residues P
Page 402 and 403: 390 Chapter 7 Conformal Mappings y
Page 406 and 407: 394 Chapter 7 Conformal Mappings π
Page 408 and 409: 396 Chapter 7 Conformal Mappings Re
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Page 412 and 413: 400 Chapter 7 Conformal Mappings De
Page 414 and 415: 402 Chapter 7 Conformal Mappings Th
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Page 418 and 419: 406 Chapter 7 Conformal Mappings Re
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Page 428 and 429: 416 Chapter 7 Conformal Mappings fo
Page 430 and 431: 418 Chapter 7 Conformal Mappings EX
Page 432 and 433: 420 Chapter 7 Conformal Mappings
Page 440 and 441: 428 Chapter 7 Conformal Mappings (a
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
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Appendix III Table of Conformal Map
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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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