Complex Analysis - Maths KU

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378 Chapter 6 Series and Residues C R s n s 2 s 4 s 3 y s1 O γ + iR γ L R γ – iR Figure 6.23 Possible contour that could be used to evaluate (7) x The limit in (7), which defines a principal value of the integral, is usually written as f(t) =� −1 {F (s)} = 1 2πi � γ+i∞ γ−i∞ e st F (s) ds, (8) where the limits of integration indicate that the integration is along the infinitely long vertical-line contour Re(s) =x = γ.Here γ is a positive real constant greater than c and greater than all the real parts of the singularities in the left half-plane.The integral in (8) is called a Bromwich contour integral.Relating (8) back to (3), we see that the kernel of the inverse transform is H(s, t) =est /2πi. The fact that F (s) has singularities s1, s2, ...,sn to the left of the line x = γ makes it possible for us to evaluate (7) by using an appropriate closed contour encircling the singularities.A closed contour C that is commonly used consists of a semicircle CR of radius R centered at (γ, 0) and a vertical line segment LR parallel to the y-axis passing through the point (γ, 0) and extending from y = γ −iR to y = γ +iR .See Figure 6.23.We take the radius R of the semicircle to be larger than the largest number in set of moduli of the singularities {|s1|, |s2|, ...,|sn|}, that is, large enough so that all the singularities lie within the semicircular region.With the contour C chosen in this manner, (7) can often be evaluated using Cauchy’s residue theorem.If we allow the radius R of the semicircle to approach ∞, the vertical part of the contour approaches the infinite vertical line that is the contour in (8). We use the contour just described in the proof of the following theorem. Theorem 6.25 Inverse Laplace Transform Suppose F (s) is a Laplace transform that has a finite number of poles s1, s2, ... , sn to the left of the vertical line Re(s) =γ and that C is the contour illustrated in Figure 6.23. If sF (s) is bounded as R →∞, then � −1 {F (s)} = n� k=1 Res � e st � F (s),sk . (9) Proof From Figure 6.23 and Cauchy’s residue theorem, we have or � e st � F (s) ds + CR � γ+iR 1 2πi γ−iR LR e st F (s) ds = e st F (s) ds =2πi n� k=1 n� k=1 Res � e st � F (s), sk Res � e st � � 1 F (s), sk − 2πi CR e st F (s) ds. (10) The �theorem is justified by letting R → ∞ in (10) and showing that lim CR R→∞ estF (s) ds = 0.Now if the semicircle CR is parametrized by
2 6.7 Applications 379 s = γ + Re iθ , π/2 ≤ θ ≤ 3π/2, then ds = Rie iθ dθ =(s − γ)idθ, and so, � 1 2πi �� 1 � � 2π � CR CR e st F (s) ds = 1 � 3π/2 e 2πi π/2 γt+Rteiθ F (γ + Re iθ )Rie iθ dθ e st � � � 3π/2 F (s) ds� 1 � � � ≤ �e 2π γt+Rteiθ�� F (γ + Re iθ ) � � � �Rie iθ��dθ. (11) π/2 To find an upper bound for the expression in (11) we examine the three moduli of the integrand of the right-hand side.First, ↓ since � �eiRt sin θ � � =1 � � �e γt+Rteiθ� � � � � = �e γt � Rt(cos θ+i sin θ) � e � = e γt e Rt cos θ . Next, for |s| sufficiently large, we can write � � Rie iθ � � = |s − γ| � �i � � ≤|s| + |γ| < |s| + |s| =2|s| and |sF (s)| 0we conclude that lim CR R→∞ estF (s) ds = 0.Finally, as R →∞we see from (10) that � γ+i∞ e st n� F (s) ds = Res � e st � F (s), sk . γ−i∞ ∗ See Problem 52 in Exercises 6.6. k=1 ✎
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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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320 Chapter 6 Series and Residues y
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322 Chapter 6 Series and Residues 2
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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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428 Chapter 7 Conformal Mappings (a
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430 Chapter 7 Conformal Mappings y
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432 Chapter 7 Conformal Mappings 3
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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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