Complex Analysis - Maths KU

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298 Chapter 5 Integration in the <strong>Complex</strong> Plane 19. If p(z) is a polynomial in z then the function f(z) =1/p(z) can be never be an entire function. 20. The function f(z) = cos z is entire and not a constant and so must be unbounded. In Problems 21–40, try to fill in the blanks without referring back to the text. 21. z(t) =e it2 , 0 ≤ t ≤ √ 2π, is a parametrization for a . 22. z(t) =z0 + e it , 0 ≤ t ≤ 2π, is a parametrization for a . 23. The difference between z1(t) =e it , 0 ≤ t ≤ 2π and z2(t) =e i(2π−t) , 0 ≤ t ≤ 2π is � . 24. (2y + x − 6ix 2 ) dz = , where C is the triangle with vertices 0, i, C 1+i, traversed counterclockwise. 25. �If f is a polynomial function and C is a simple closed contour, then f(z) dz = . 26. 27. 28. 29. 30. � � � � � C C C C C C z Im(z) dz = , where C is given by z(t) =2t + t 2 i, 0 ≤ t ≤ 1. |z| 2 dz = , where C is the line segment for 1 − i to 1 + i. (¯z) n dz = , where n is an integer and C is z(t) =e it , 0 ≤ t ≤ 2π. sin z 2 dz = , where C is given by z(t) =2i +4eit , 0 ≤ t ≤ π/2. sec zdz= , where C is |z| =1. � 1 31. C z(z − 1) � 32. If f(z) = C dz = , where C is |z − 1| = 1 2 . ξ 2 +6ξ − 2 ξ − z dξ, where C is |z| = 3, then f(1 + i) = . 33. If f(z) = z 3 + e z and C is a contour z = 8e it � , 0 ≤ t ≤ 2π, then f(z) dz = . C (z + πi) 3 �� 34. If | f(z) |≤2 on the circle |z| =3, then � � f(z) dz � � ≤ . 35. �If n is a positive integer and C is the contour |z| = 2, then z −n e z dz = . C � cos z 36. On |z| = 1, the contour integral dz equals C zn for n = 2, and equals for n =3. for n = 1, equals 37. � z C n ⎧ ⎨ 0, dz = ⎩ 2πi, |z| =1. if n if n , where n is an integer and C is the circle C
Chapter 5 Review Quiz 299 � z 38. The value of the integral dz on the contour C shown in Figure 5.59 is C z + i . � 39. The value of the integral (2z +1)dz on the contour C, comprised of line C segments C1, C2, ... , C11 shown in Figure 5.60 is . � e 40. The value of the integral C z z2 dz on the closed contour C shown in (z − πi) Figure 5.61 is . –2 C y Figure 5.59 Figure for Problem 38 (–2, 3) C 7 (–2, 1) C 6 C 8 (0, 2) 3 y (0, 5) C5 (3, 5) C 9 C 11 (1, 2) C 10 (1, 1) x (3, 3) 0 C 3 1 Figure 5.60 Figure for Problem 39 1 y 1 Figure 5.61 Figure for Problem 40 C C 2 C 3 x C 4 (5, 3) x
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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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348 Chapter 6 Series and Residues E
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350 Chapter 6 Series and Residues 4
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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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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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370 Chapter 6 Series and Residues E
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374 Chapter 6 Series and Residues I
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376 Chapter 6 Series and Residues y
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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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384 Chapter 6 Series and Residues (
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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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