Complex Analysis - Maths KU

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APP-4 Appendix II Proof of the Cauchy-Goursat Theorem Note ☞ 7.2 Appendix II Proof of the Cauchy-Goursat Theorem In Section 5.3 we proved Cauchy’s theorem using Green’s theorem. This simple proof was possible because of the hypothesis of continuity of f ′ throughout a simply connected domain D. The French mathematician Edouard Goursat (1858–1936) published a proof of Cauchy’s theorem in 1900 without this continuity assumption. As a result his name was thereafter linked with Cauchy’s in the title of one of the most fundamental of all theorems in complex analysis. In this appendix we discuss how the full proof of the Cauchy-Goursat theorem is accomplished. To avoid needless repetition throughout the following discussion we will take for granted that we are working in a simply connected domain D and that f represents a complex function analytic in D. The proof of the Cauchy-Goursat theorem is accomplished in three steps. The first two steps are helping theorems—sometimes called lemmas—which are actually special cases of the Cauchy-Goursat theorem. The first of these helping theorems deals with integrals along a triangular contour and the second deals with integrals along a closed polygonal contour. The first theorem is used in the proof of the second, and the second theorem is utilized to establish the Cauchy-Goursat theorem in its full generality. We will prove the first theorem, but because of its length, we will simply sketch the proofs of the remaining two. Theorem A.2 Triangular Contour If ∆ is a triangular contour lying entirely within D, then � ∆ f(z) dz =0. Proof Let ∆ be the triangular contour shown in Figure AII.1; the vertices of ∆ are labeled A, B, and C. We form four smaller triangles C1, C2, C3, and C4 by joining the midpoints E, F , and G of the sides of ∆ by straight line ∆ B C Figure AII.1 Triangular contour ∆ within D A C 1 ∆ C 4 B F C A C 3 E G C 2 Figure AII.2 Triangular contours C1, C2, C3, and C4
Appendix I Proof of Theorem 2.1 APP-3 In particular, if |f(z) − L| 0 such that |u(x, y) − u0| 0. z→z0 Therefore, from our assumption that lim (x,y)→(x0,y0) u(x, y) = u0 and (7) of Section 2.6, we have that there is a δ1 > 0 such that |u(x, y) − u0|
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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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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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392 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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Page 432 and 433: 420 Chapter 7 Conformal Mappings
Page 440 and 441: 428 Chapter 7 Conformal Mappings (a
Page 444 and 445: 432 Chapter 7 Conformal Mappings 3
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: Integrals � � � FE , GF , EG
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 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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