Complex Analysis - Maths KU

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132 Chapter 2 <strong>Complex</strong> Functions and Mappings 52. If f is a function for which lim x→0 f(x + i0) = 0 and lim y→0 f(0 + iy) = 0, then can you conclude that lim z→0 f(z) = 0? Explain. 53. (a) Prove that the function f(z) = Arg(z) is discontinuous at every point on the negative real axis. (b) Prove that the function f1 defined by f1(z) =θ, −π
–4 –4 –3 –2 (–2, –1) 4 3 2 1 y –1 1 2 3 4 –1 (1, –1) –2 –3 –4 (1, 2) (a) Values of F plotted as position vectors –3 (–2, 1) –2 4 3 y 2 (0, 1) 1 –1 1 2 3 4 –1 –2 –3 –4 (1, 1) (1, –2) (b) Values of F plotted with initial point at (x, y) x x 2.7 Applications 133 Vector Fields In multivariable calculus, a vector-valued function of two real variables F(x, y) =(P (x, y),Q(x, y)) (1) is also called a two-dimensional vector field. Using the standard orthogonal unit basis vectors i and j, we can also express the vector field in (1) as: F(x, y) =P (x, y)i + Q(x, y)j. (2) For example, the function F(x, y) =(x + y)i +(2xy)j is a two-dimensional vector field, for which, say, F(1, 3) = (1 + 3)i +(2· 1 · 3)j =4i +6j. Values of a function F given by(2) are vectors that can be plotted as position vectors with initial point at the origin. However, in order to obtain a graphical representation of the vector field (2) that displays the relation between the input (x, y) and the output F(x, y), we plot the vector F(x, y) with initial point (x, y) and terminal point (x + P (x, y), y+ Q(x, y)). For example, in Figure 2.57(a) the four functional values F(1, 1) = i − j, F(0, 1) = −j, F(1, −2) = i+2j, and F(−2, 1) = −2i−j of the vector field F(x, y) =xi−yj are plotted as position vectors, whereas in Figure 2.57(b) we have a portion of the graphical representation of the vector field obtained byplotting these four vectors with initial point at (x, y). Specifically, Figure 2.57(b) consists of the four vectors i − j, −j, i +2j, and −2i − j plotted with initial points at (1, 1), (0, 1), (1, −2), and (−2, 1), and terminal points (2, 0), (0, 0), (2, 0), and (−4, 0), respectively. Figure 2.57 Some vector values of the function F(x, y) =xi − yj <strong>Complex</strong> Functions as Vector Fields There is a natural wayto y represent a vector field F(x, y) =P (x, y)i+Q(x, y)j with a complex function f. Namely, we use the functions P and Q as the real and imaginaryparts of f, in which case, we saythat the complex function f(z) =P (x, y)+iQ(x, y) is the complex representation of the vector field F(x, y) =P (x, y)i + 4 Q(x, y)j. Conversely, any complex function f(z) =u(x, y)+iv(x, y) has an associated vector field F(x, y) =u(x, y)i+v(x, y)j. From this point on we shall 2 refer to both F(x, y) =P (x, y)i + Q(x, y)j and f(x, y) =u(x, y)+iv(x, y) –6 –4 –2 2 4 x as vector fields. As an example of this discussion, consider the vector field f(z) =¯z. Since f(z) =x − iy, the function f is the complex representation –2 of the vector field F(x, y) =xi − yj. Part of this vector field was shown in Figure 2.57(b). A more complete representation of the vector field f(z) =¯z –4 is shown in Figure 2.58 (this plot was created in Mathematica). Observe that the vector field plotted in Figure 2.58 gives a graphical representation of the complex function f(z) =¯z that is different from a mapping. Compare with Figure 2.58 The vector field f(z) =z Figure 2.41 in Section 2.5.
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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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Page 173 and 174: 3.3 Harmonic Functions 161 EXAMPLE
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Page 177 and 178: 3.4 Applications 165 tangent is the
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182 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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