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List of Tables, Notes, Diagrams, Classifications, and Lists |
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xvii | |
| Series Preface |
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xxi | |
| Preface |
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xxv | |
| About the Author |
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xxvii | |
| Acknowledgments |
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xxix | |
| Mathematical Symbols |
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xxxi | |
| Physical Quantities |
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xxxvii | |
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Part 1 Complex Domain: Circuits and Stability |
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1 | (120) |
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1 Complex Numbers and Quaternions |
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3 | (10) |
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1.1 Peano (1889, 1891) Postulates for Natural Numbers |
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3 | (2) |
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1.2 Irrational Numbers (Pythagoras, VI b.c.) and Dedekind (1858) Section |
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5 | (1) |
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1.3 Cartesian Parts: Real and Imaginary (Argand, 1806; Descartes, 1637a; Gauss, 1797) |
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6 | (1) |
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1.4 Polar Coordinates: Modulus and Argument |
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6 | (1) |
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1.5 Moivre's Formula, Origin and Infinity |
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7 | (1) |
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1.6 Conjugate and Reflection on the Origin and Axis |
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8 | (1) |
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1.7 Power with Integral Exponent and Logarithm |
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9 | (1) |
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1.8 Real, Imaginary, and Complex Exponential |
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9 | (1) |
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1.9 Noncommutative Product of Quaternions (Hamilton, 1843) |
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10 | (3) |
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2 Stability of an Equilibrium Position |
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13 | (10) |
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2.1 Trajectory Following a Perturbation of Equilibrium |
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13 | (1) |
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2.2 Oscillatory Motion with Constant Amplitude |
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14 | (1) |
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2.3 Attenuation or Amplification and Stability or Instability |
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15 | (1) |
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2.4 Damped Oscillation or Overstable Growth |
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15 | (1) |
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2.5 General Relations for Amplitudes and Phases |
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16 | (1) |
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2.6 Predominantly or Weakly Oscillatory Motion |
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16 | (1) |
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2.7 Frequency and Attenuation/Amplification Factor |
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17 | (1) |
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2.8 Differential Equation and Stability Criteria |
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18 | (1) |
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2.9 Initial Conditions for Harmonic Oscillator |
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19 | (4) |
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3 Addition, Product, and Inverses |
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23 | (10) |
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3.1 Complex Addition and Rule of the Parallelogram |
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23 | (1) |
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3.2 Modulus, Argument, and Triangular Equalities (Pythagoras, VI b.c.) |
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24 | (1) |
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3.3 Complex Product, Homothety, and Rotation |
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25 | (1) |
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3.4 Meaning of the Imaginary Symbol "i" |
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26 | (1) |
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3.5 Conjugate of the Sum, Product, and Inversion |
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27 | (1) |
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3.6 Complex Representation of Real Quantities |
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27 | (1) |
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3.7 Trigonometric Addition and Multiplication Formulas |
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28 | (2) |
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3.8 Conjugate Complex and Triangular Inequalities |
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30 | (1) |
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3.9 Generalized Schwartz (1890) or Polygonal Inequality |
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30 | (3) |
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4 Impedance of Associations of Circuits |
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33 | (10) |
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4.1 Inertia, Friction, and Elastic Forces |
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33 | (1) |
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4.2 Free and Forced Motion of Circuit |
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34 | (1) |
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4.3 Electrical Induction, Resistance, and Capacity |
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35 | (1) |
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4.4 Decomposition of Impedance into Inductance and Reactance |
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36 | (1) |
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4.5 Activity in Terms of the Velocity, Force, and Impedance |
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36 | (2) |
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4.6 Mechanical Circuits in Parallel or Series |
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38 | (1) |
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4.7 Electromechanical Analogy and Contrasting Laws |
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39 | (1) |
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4.8 Comparison of Two Circuits in Parallel and in Series |
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40 | (1) |
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4.9 Hybrid Associations of Three Circuits |
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40 | (3) |
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5 Power, Root, and Logarithm |
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43 | (10) |
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5.1 n-th Power as the (n - 1)- Times Iterated Product |
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43 | (1) |
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5.2 Discrete Set of Points on Logarithmic Spiral |
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44 | (1) |
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5.3 Inversion of the Power: Roots of Order n |
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45 | (1) |
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5.4 Regular Polygon Contained in a Circle |
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45 | (2) |
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5.5 Multiple Sums of Sines or Cosines of Equal Angles |
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47 | (1) |
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5.6 Single-, Multi-, and Many- Valued Functions |
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48 | (1) |
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5.7 Power with Complex Base and Exponent |
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49 | (1) |
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5.8 Limiting Behavior at the Origin and Infinity |
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49 | (1) |
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5.9 Vanishing and Divergence on Alternate Sectors |
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50 | (3) |
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6 Electron in an Electromagnetic Field |
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53 | (12) |
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6.1 Electromagnetic or Laplace-Lorentz Force |
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53 | (1) |
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6.2 Uniform Fields and Larmor (1897) Frequency |
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54 | (1) |
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6.3 Longitudinal Translation and Transverse Rotation |
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54 | (1) |
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6.4 Components of the Velocity and Trajectory of the Particle |
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55 | (1) |
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6.5 Linear, Circular, and Helical Motion |
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55 | (2) |
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6.6 Linear Acceleration and Cycloid in the Plane |
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57 | (2) |
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6.7 Elongated Helix and Plane Trochoid |
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59 | (1) |
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6.8 Oval and Conical Helices and Magnetic Focusing |
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60 | (2) |
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6.9 Separation of Isotopes in a Mass Spectrograph |
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62 | (3) |
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7 Multivalued Functions, Branch-Points, and Branch-Cuts |
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65 | (16) |
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7.1 Riemann (1857) Surface of a Multivalued Function |
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65 | (2) |
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7.2 Denumerable Infinity of Connected Sheets |
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67 | (2) |
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7.3 Principal Branch in the Cut-Plane |
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69 | (1) |
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7.4 Jump Discontinuity across a Branch-Cut |
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70 | (1) |
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7.5 Semiinfinite Cut Joining a Branch-Point to Infinity |
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70 | (1) |
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7.6 Infinite Derivative of a Function at a Branch-Point |
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71 | (1) |
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7.7 Theorem and Method for the Identification of Branch-Points |
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72 | (1) |
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7.8 Elementary Functions with Two Branch-Points |
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73 | (4) |
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7.9 Functions with Several Branch-Points and Branch-Cuts |
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77 | (4) |
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8 Motion of a Pendulum and a Ship |
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81 | (10) |
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8.1 Stability of a Suspended or Inverted Pendulum |
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81 | (2) |
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8.2 Motion of the Pendulum and Force along the Rod |
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83 | (1) |
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8.3 Metacentric Distance and Rolling Torque |
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84 | (1) |
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8.4 Length of the Pendulum Equivalent to the Ship |
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84 | (1) |
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8.5 Kinetic, Potential, and Total Energy |
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85 | (1) |
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8.6 Linearization in the Vicinity of the Equilibrium Position |
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86 | (1) |
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8.7 Branch-Points as Boundaries of the Angular Motion |
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87 | (1) |
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8.8 Oscilation about a Position of Stable Equilibrium (Galileo, 1583) |
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88 | (1) |
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8.9 Divergence Away from the Position of Unstable Equilibrium |
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89 | (2) |
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9 Stereographic Projection and Genus of a Surface |
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91 | (12) |
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9.1 Injective, Surjective, and Bijective Mappings |
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91 | (2) |
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9.2 Unit Sphere and the Complex Plane |
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93 | (1) |
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9.3 Direct and Inverse Stereographic Transformation |
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94 | (1) |
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9.4 Mapping of a Circle onto a Straight Line |
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95 | (1) |
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9.5 Projection of a Circle into Another Circle |
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95 | (1) |
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9.6 Isomorphism of Circle and Real Line |
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96 | (1) |
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9.7 Continuous Deformation and Topological Sphere |
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97 | (1) |
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9.8 Sphere with One Handle and Toroidal Topology |
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98 | (1) |
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9.9 Torus with Holes or Sphere with Handles |
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99 | (4) |
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10 Examples 10.1 to 10.20 |
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103 | (18) |
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Part 2 Integrals and Residues: Flows and Gravity |
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121 | (174) |
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11 Differentiation and Holomorphic Functions |
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123 | (12) |
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11.1 Function, Neighborhood, Limit, and Uniformity |
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123 | (1) |
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11.2 Continuity, Incremental Ratio, and Derivate (Newton, 1670; Leibnitz, 1684) |
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124 | (1) |
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11.3 Holomorphic Function and Continuous Derivatives |
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125 | (3) |
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11.4 Cauchy (1821)---Riemann (1851) Conditions in Cartesian and Polar Coordinates |
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128 | (1) |
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11.5 Formulas for the Derivative and its modulus and Argument |
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128 | (1) |
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11.6 Cartesian and Polar Laplace (1825) Equation |
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129 | (1) |
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11.7 Gradient, Divergence, Curl, and Laplacian |
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130 | (1) |
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11.8 Families of Plane Orthogonal Curves |
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131 | (1) |
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11.9 Orthogonal Plane Curvilinear Coordinates |
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132 | (3) |
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12 Potential Flow and Multipoles |
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135 | (16) |
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12.1 Circulation, Potential, Curl, and Vorticity |
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135 | (2) |
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12.2 Flow Rate, Stream Function, Divergence, and Dilatation (Lagrange, 1781; Rankine, 1864) |
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137 | (1) |
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12.3 Complex Potential and Conjugate Velocity |
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138 | (1) |
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12.4 Irrotational Flow due to a Source or Sink |
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139 | (2) |
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12.5 Incompressible Flow due to a Vortex |
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141 | (1) |
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12.6 Superposition as a Monopole and Spiral Flow |
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142 | (1) |
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12.7 Dipole as the Limit of Two Opposing Monopoles |
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142 | (3) |
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12.8 Quadrupole Moment and Rule of Differentiation |
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145 | (2) |
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12.9 Arbitrary Multipole and Directivity Lobes |
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147 | (4) |
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13 Primitive and Contour Integrals |
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151 | (12) |
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13.1 Existence and Properties of the Primitive of a Function |
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151 | (1) |
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13.2 Riemann Integral of a Complex Function (Cauchy, 1825; Riemann, 1851) |
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152 | (2) |
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13.3 Rectifiable Curves and Bounded Functions |
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154 | (1) |
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13.4 Parametric Limits and Uniform Continuity |
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155 | (1) |
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13.5 Complex Loop and Contour Integrals |
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156 | (1) |
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13.6 Reciprocal Theorems of Cauchy (1825) and Morera (1886)---Osgood (1896) |
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157 | (1) |
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13.7 Integration by Parts and Chain Rule (Leibnitz, 1864) |
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158 | (1) |
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13.8 Derivation of an Integral with Regard to a Parameter |
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159 | (1) |
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13.9 Parametric Integral with Variable End-Points |
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160 | (3) |
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14 Pressure and Corner Flows |
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163 | (24) |
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14.1 Mass Conservation and Equation of Continuity |
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163 | (2) |
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14.2 Inviscid Momentum Equation (Euler, 1752, 1759) |
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165 | (1) |
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14.3 Adiabatic Condition and Equation of State |
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166 | (1) |
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14.4 Homentropic Flow and Conservation of Circulation (Helmholtz, 1858; Kelvin, 1869) |
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167 | (1) |
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14.5 Hydrostatic, Dynamic, and Stagnation Pressures (Torricelli, 1643; Bernoulli, 1738) |
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168 | (1) |
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14.6 Compressibility Effects and the Pitot Tube (1732) |
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169 | (4) |
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14.7 Venturi Tube (Herschel, 1887) and Variable-Area Duct |
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173 | (4) |
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14.8 Corner Flows and Multipoles at Infinity |
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177 | (4) |
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14.9 Stream past a Wedge and Sharp Edge |
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181 | (6) |
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15 Loop Integrals and Poles |
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187 | (14) |
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15.1 Cauchy (1821) First Theorem on Integrals |
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187 | (1) |
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15.2 Doubly-Connected Region and "Shrinking" of a Loop |
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188 | (1) |
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15.3 Second Cauchy (1821) Theorem: Value of the Function |
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189 | (1) |
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15.4 Third Cauchy (1821) Theorem: All the Derivates |
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190 | (1) |
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15.5 Inclusion or Exclusion of Singularities on the Boundary |
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191 | (2) |
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15.6 Holomorphic Function in a Multiply-Connected Region |
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193 | (1) |
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15.7 Residue of a Function at a Simple Pole |
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194 | (2) |
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15.8 Multiple Pole or Pole of Order n |
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196 | (2) |
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15.9 Loop Integral with Poles in the Interior and on the Boundary |
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198 | (3) |
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201 | (24) |
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16.1 Identical Image on a Rigid Wall (Rankine, 1864) |
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201 | (5) |
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16.2 Image Vortex with Opposite Circulation |
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206 | (3) |
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16.3 Effect of Wall on Monopole or Spiral Source |
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209 | (1) |
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16.4 Far-Field of Multipole near a Hard Wall |
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210 | (1) |
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16.5 Monopole in a Hard-Walled Rectangular Corner |
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211 | (1) |
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16.6 Trajectories of a Vortex or Source/Sink in a Corner |
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212 | (3) |
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16.7 Flow and Forces for a Monopole in a Rectangular Corner (Grobli, 1877; Greenhill, 1878) |
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215 | (5) |
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16.8 Multiple Identical Source/Sink Images |
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220 | (1) |
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16.9 Alternating Vortices in a Rigid Corner |
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221 | (4) |
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17 Improper Integrals and Principal Value |
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225 | (18) |
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17.1 Improper Uni(bi)lateral Integrals of the Three Kinds |
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225 | (1) |
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17.2 Transformation of a Straight Segment into a Circle |
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226 | (2) |
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17.3 Closing a Straight Line by a Half-Circle |
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228 | (1) |
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17.4 Connecting the Real Axis in the Upper/Lower Half-plane |
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229 | (2) |
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17.5 Integrals with an Oscillating Factor (Jordan, 1894) |
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231 | (3) |
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17.6 The Localization Lemma for Holomorphic Functions |
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234 | (1) |
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17.7 Surrounding a Semiinfinite Branch-Cut |
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235 | (3) |
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17.8 Branch-Point within the Path of Integration |
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238 | (1) |
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17.9 Cauchy (1821) Principal Value of an Integral |
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239 | (4) |
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18 Mass and the Gravity Field |
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243 | (18) |
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18.1 Irrotational Flow due to Sources or Sinks |
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243 | (2) |
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18.2 Incompressible Flow due to a Vorticity Distribution |
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245 | (1) |
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18.3 Gravity Field and Gravitational Constant |
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246 | (1) |
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18.4 Line, Surface, and Volume Mass Distribution |
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247 | (1) |
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18.5 Gravity Force of Attraction (Newton, 1687) |
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248 | (1) |
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18.6 Gravity Field of a Homogeneous Slab |
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249 | (2) |
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18.7 Gravity Field inside and outside the Mass |
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251 | (2) |
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18.8 Field due to a Distribution of Infinite Extent |
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253 | (3) |
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18.9 Multipolar Representation of the Gravity Field |
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256 | (5) |
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19 Cauchy Conditions and Infinitesimals |
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261 | (12) |
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19.1 Calculation of Riemann Integrals Using the Definition |
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261 | (1) |
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19.2 Mean Value Theorem and Bounds |
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262 | (1) |
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19.3 Division into Internal and Boundary Regions |
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262 | (2) |
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19.4 Function Holomorphic in the Interior and on the Boundary |
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264 | (1) |
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19.5 Uniform Continuity on the Boundary (Goursat, 1900) |
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265 | (1) |
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19.6 Isolated Ignorable Singularities on the Boundary (Littlewood, 1944) |
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265 | (1) |
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19.7 Infinitesimals of the Same or Higher Order |
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266 | (1) |
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19.8 Zero of Order n and L'Hospital's (1696; Bernoulli, 1691) Rule |
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267 | (2) |
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19.9 Calculation of the Residues of Ratios of Functions |
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269 | (4) |
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20 Examples 20.1 to 20.20 |
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273 | (22) |
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Part 3 Power Series: Electricity and Magnetism |
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295 | (246) |
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21 Convergence of and Operations on Series |
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297 | (20) |
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21.1 Convergent, Divergent, and Oscillating Series |
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297 | (3) |
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21.2 Association of Terms and Sum of a Series |
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300 | (1) |
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21.3 Absolute and Conditional Convergence (Dirichlet, 1837) |
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301 | (2) |
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21.4 Permutation of Terms and Product of Series (Cauchy, 1821) |
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303 | (1) |
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21.5 Uniform Convergence and Series of Functions |
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304 | (4) |
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21.6 Limit, Differentiation, and Integration Term-by-Term |
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308 | (2) |
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21.7 Total Convergence and Weierstrass M-Test (1876) |
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310 | (1) |
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21.8 Geometric, Logarithmic, and Inverse-Power Series |
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311 | (2) |
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21.9 Convergence inside, outside, and on the Unit Circle |
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313 | (4) |
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22 Multiple Reflections in a Lens |
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317 | (18) |
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22.1 Period, Frequency, Wavelength, and Wavevector |
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317 | (2) |
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22.2 Reflection, Transmission, and Inaccessible Regions (Snell, 1626; Descartes, 1637b; Fresnel, 1823) |
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319 | (2) |
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22.3 Wave Scattering and Fastest Path (Fermat, 1657) |
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321 | (2) |
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22.4 Reflection and Transmission of Acoustic Waves |
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323 | (3) |
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22.5 Adsorption at an Interface and Internal Absorption |
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326 | (2) |
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22.6 Multiple Reflections between Parallel Interfaces |
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328 | (1) |
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22.7 Total Reflection, Transmission, and Damping Coefficients |
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329 | (1) |
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22.8 Multiple Media, Transparency, and Opaqueness |
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330 | (1) |
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22.9 Constructive and Destructive Interference (Bragg, 1912) |
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331 | (4) |
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23 Analytic Series of Ascending Powers |
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335 | (18) |
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23.1 Harmonic Function and Mean Value on a Circle |
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335 | (1) |
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23.2 Lemmas of Constancy and Maximum Modulus |
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336 | (1) |
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23.3 Monotonic Chain of Regions and Loops |
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337 | (1) |
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23.4 Geometric Series of Holomorphic Functions |
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338 | (1) |
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23.5 Regions of Absolute and Uniform Convergence |
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339 | (1) |
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23.6 Lagrange (1770)---Burmann (1799) Series and Implicit Derivation |
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340 | (1) |
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23.7 Taylor (1715) and Stirling (1717)---Maclaurin (1742) Theorems |
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341 | (1) |
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23.8 Implicit Derivatives and Mean-Value Theorem |
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342 | (1) |
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23.9 Darboux Expansion (1876) and Lagrange/Cauchy Remainders |
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343 | (10) |
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24 Electrostatics, Charges, and Dielectrics |
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353 | (22) |
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24.1 Electric Field, Displacement, and Polarization (Maxwell, 1873) |
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353 | (1) |
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24.2 Dielectric Permittivity and Electric Susceptibility |
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354 | (1) |
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24.3 Potential due to Charges and Electric Force (Coulomb, 1785) |
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355 | (1) |
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24.4 Multipole near Insulating or Conducting Wall |
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356 | (3) |
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24.5 Identical or Alternating Images in a Corner |
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359 | (2) |
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24.6 Cylinder in a Uniform Electric Field |
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361 | (2) |
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24.7 Reciprocal Point and First Circle Theorem (Kirchhoff, 1845) |
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363 | (2) |
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24.8 Induced Electric Charges on a Cylinder |
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365 | (3) |
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24.9 Charge near Interface between Two Dielectrics |
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368 | (7) |
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25 Singular Series of Ascending-Descending Powers |
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375 | (14) |
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25.1 Lemma of the Extrema and Doubly-Connected Chain |
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375 | (3) |
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25.2 Ascending and Descending Geometric Series |
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378 | (1) |
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25.3 Total Convergence in a Closed Subregion |
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379 | (1) |
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25.4 Absolute Convergence in an Open Region |
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380 | (1) |
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25.5 Series of Teixeira (1900): Coefficients and Remainder |
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380 | (1) |
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25.6 Restriction to Laurent (1843)---Weierstress (1841) and Laurent---Maclaurin Series |
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381 | (1) |
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25.7 Hierarchy of Power Series Expansions |
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382 | (1) |
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25.8 Coefficients of Reversion of Series of Third-Order |
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383 | (1) |
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25.9 Binomial Expansion and Series and Inverse Powers |
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384 | (5) |
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26 Magnetostatics, Currents, and Permeability |
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389 | (24) |
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26.1 Magnetic Field, Induction, and Polarization (Maxwell, 1873) |
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389 | (1) |
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26.2 Magnetic Permeability, Susceptibility, and Field Function |
|
|
390 | (1) |
|
26.3 Electric Current and Magnetic Force (Biot-Savart) |
|
|
391 | (1) |
|
26.4 Hydrodynamic, Electromagnetic, and Gravity Multipoles |
|
|
392 | (2) |
|
26.5 Current near Conducting or Insulating Plane |
|
|
394 | (2) |
|
26.6 Image Electric Currents in a Corner |
|
|
396 | (1) |
|
26.7 Cylinder in a Magnetic Field or near a Line-Current |
|
|
397 | (3) |
|
26.8 Current near Cylindrical Magnetic Interface |
|
|
400 | (7) |
|
26.9 Infinite Magnetic Dipole Distribution |
|
|
407 | (6) |
|
27 Classification of Singularities and Functions |
|
|
413 | (26) |
|
27.1 Chain of Inclusion of Real Functions |
|
|
413 | (3) |
|
27.2 Set of Coincidences for Complex Functions |
|
|
416 | (2) |
|
27.3 Ordinary Points, Zeros, and Singularities |
|
|
418 | (2) |
|
27.4 Residues at Poles and Essential Singularities |
|
|
420 | (1) |
|
27.5 Inversion of the Origin and singularity at Infinity |
|
|
421 | (2) |
|
27.6 Identification of Constants (Cauchy, 1844; Liouville, 1847) |
|
|
423 | (1) |
|
27.7 Definition of Polynomial and Rational Functions |
|
|
424 | (2) |
|
27.8 Essential Singularity as an Accumulation of Poles |
|
|
426 | (2) |
|
27.9 Integral, Meromorphic, and Polymorphic Functions |
|
|
428 | (11) |
|
28 Forces and Moments on Bodies |
|
|
439 | (54) |
|
28.1 Kinetic, Electric, Magnetic, and Gravity Energies |
|
|
439 | (3) |
|
28.2 Drag/Thrust, Lift/Downforce, and Pitching Moment (Kutta, 1902a; Joukowski, 1906; Blasius, 1910) |
|
|
442 | (11) |
|
28.3 Hydrodynamic, Electromagnetic, and Gravity Forces |
|
|
453 | (7) |
|
28.4 Fairing due to a Source or Sink in a Stream (Rankine, 1871) |
|
|
460 | (2) |
|
28.5 Oval/Valley/Throat due to a Source and Sink Pair |
|
|
462 | (4) |
|
28.6 Virtual Mass of a Cylinder and Cavitation |
|
|
466 | (6) |
|
28.7 Flow past a Cylinder with Circulation |
|
|
472 | (7) |
|
28.8 Moving Vortex and Source/Sink Image System |
|
|
479 | (6) |
|
28.9 Dipole outside or inside a Cylinder |
|
|
485 | (8) |
|
29 Combined Test of Convergence |
|
|
493 | (22) |
|
29.1 Behavior of Series at all Points of the Complex Plane |
|
|
493 | (5) |
|
29.2 Cauchy (1821) Necessary and Sufficient Conditions |
|
|
498 | (3) |
|
29.3 Region of Convergence and D'Alembert's Ratio (1768) |
|
|
501 | (1) |
|
29.4 Convergence of Integrals and Harmonic Series |
|
|
502 | (2) |
|
29.5 Gauss Test (1812a) and Euler (1735)-Mascheroni (1790) Constant |
|
|
504 | (2) |
|
29.6 Criteria and Sums of Abel (1826, 1839)-Dirichlet (1862) |
|
|
506 | (2) |
|
29.7 Boundary of Convergence and Weierstrass K-Test (1856) |
|
|
508 | (2) |
|
29.8 Radius and Exponent of a Power Series |
|
|
510 | (1) |
|
29.9 Gaussian or Three-Parameter Hypergeometric Series (Gauss, 1812b) |
|
|
511 | (4) |
|
30 Examples 30.1 to 30.20 |
|
|
515 | (26) |
|
Part 4 Conformal Mapping: Heat and Aerodynamics |
|
|
541 | (412) |
|
31 Analytic Continuation and Rational Functions |
|
|
543 | (24) |
|
31.1 Theorem of Monodromy and Lacunary Functions (Osgood, 1929) |
|
|
543 | (4) |
|
31.2 Conjugate Property and Reflection Principle (Riemann, 1863; Schwartz, 1890) |
|
|
547 | (3) |
|
31.3 Analytic Extension with Jump across an Arc (Plemelj, 1908) |
|
|
550 | (3) |
|
31.4 The Cauchy (1821) Fourth Integral Theorem |
|
|
553 | (1) |
|
31.5 Number of Zeros and Poles of a Function |
|
|
554 | (1) |
|
31.6 Theorem of Rouche (1858) and Fundamental Theorem of Algebra |
|
|
555 | (3) |
|
31.7 Legendre's Theorem and Roots of Polynomials |
|
|
558 | (2) |
|
31.8 Rational Functions and Simple Fractions |
|
|
560 | (3) |
|
31.9 Decomposition into Partial Fractions and Rational Integrals |
|
|
563 | (4) |
|
32 Steady Heat Conduction |
|
|
567 | (28) |
|
32.1 Heat Flux and Thermal Conductivity (Fourier, 1818) |
|
|
567 | (3) |
|
32.2 Regularity, Asymptotic, and Boundary Conditions |
|
|
570 | (1) |
|
32.3 Irrotational and Solenoidal Potential Fields |
|
|
571 | (1) |
|
32.4 Corner with Isothermal or Adiabatic Walls |
|
|
572 | (2) |
|
32.5 Solid Cylinder and Cylindrical Cavity |
|
|
574 | (4) |
|
32.6 Hollow Tube with Thick or Thin Walls |
|
|
578 | (4) |
|
32.7 Convective Transfer in Heat Exchangers |
|
|
582 | (4) |
|
32.8 Concentric Cylinders of Different Materials |
|
|
586 | (3) |
|
32.9 Parallel Walls of an Inhomogeneous Substance |
|
|
589 | (6) |
|
33 Conformal and Critical Points |
|
|
595 | (20) |
|
33.1 Preservation of Modulus and Direction of Angles |
|
|
595 | (1) |
|
33.2 Inversion of Angles and Isogonal Mapping |
|
|
596 | (1) |
|
33.3 Transformation of Angles, Lengths, and Areas |
|
|
597 | (3) |
|
33.4 Critical Points of the First and Second Kinds |
|
|
600 | (2) |
|
33.5 Multiplication and Division of Angles into Edges |
|
|
602 | (2) |
|
33.6 Interior Polygonal Transformation (Christoffel, 1868; Schwartz, 1868) |
|
|
604 | (3) |
|
33.7 Interior and Exterior Mappings and Point-at-Infinity |
|
|
607 | (2) |
|
33.8 Mapping of a Disk into the Interior of a Polygon |
|
|
609 | (1) |
|
33.9 Finite Interior and Overlapping Exterior |
|
|
610 | (5) |
|
34 Wing Sections and Planforms |
|
|
615 | (54) |
|
34.1 Flow Past a Flat Plate and Kutta (1902b) Condition |
|
|
616 | (7) |
|
34.2 Joukowski (1910) Transformation and the Elliptic Cylinder |
|
|
623 | (5) |
|
34.3 Circular Arc and Symmetric Airfoils |
|
|
628 | (4) |
|
34.4 Cambered or Unsymmetric Joukowski (1916) Airfoil |
|
|
632 | (5) |
|
34.5 Parametric Families and Generic Airfoils (von Karman-Trefftz, 1918; von Mises, 1920; Carafoli) |
|
|
637 | (6) |
|
34.6 Lift and Pitching Moment Axis and Coefficients |
|
|
643 | (8) |
|
34.7 Spanwise Distribution of Circulation along a Lifting-Line (Prandtl, 1918) |
|
|
651 | (9) |
|
34.8 Uniform Downwash and Elliptic Loading |
|
|
660 | (3) |
|
34.9 Induced, Form, and Total Drag |
|
|
663 | (6) |
|
35 Linear and Homographic Transformations |
|
|
669 | (20) |
|
35.1 Rotation, Translation, and Isometric Mappings |
|
|
669 | (2) |
|
35.2 Group of Linear Mappings and Homothety |
|
|
671 | (1) |
|
35.3 Attractive, Repulsive, and Indifferent Limit Points |
|
|
672 | (1) |
|
35.4 Univalent Mapping and Homographic Transformation |
|
|
673 | (2) |
|
35.5 Bilinear Group (Mobius) and Self-Inverse Function |
|
|
675 | (2) |
|
35.6 Four-Point Cross-Ratio and Fixed Points |
|
|
677 | (1) |
|
35.7 Reciprocal Points with Regard to the Circle and the Straight Line |
|
|
678 | (5) |
|
35.8 Mapping of a Half-Plane into a Unit Disk |
|
|
683 | (1) |
|
35.9 Mapping between Interiors and Exteriors of Circles |
|
|
684 | (5) |
|
36 Channels, Condenses, and Wakes |
|
|
689 | (66) |
|
36.1 Rounded Wedge and Cylindrical Indentation |
|
|
690 | (7) |
|
36.2 Identical/Alternating Images for Irrotational/Solenoidal Fields |
|
|
697 | (7) |
|
36.3 Path of a Monopole past a Sharp Edge |
|
|
704 | (9) |
|
36.4 Circulation around a flat Plate and Flow through a Slit |
|
|
713 | (7) |
|
36.5 Convergent Channel (Harris, 1901) and Added Length |
|
|
720 | (7) |
|
36.6 Monopole Images on Parallel Walls |
|
|
727 | (4) |
|
36.7 Confined Vortex and Single Vortex row |
|
|
731 | (6) |
|
36.8 Sourc/Sink in a Well, on a Wall or at a Corner |
|
|
737 | (6) |
|
36.9 Parallel and Staggered Double Vortex Street (von karman, 1911; Lamb, 1932) |
|
|
743 | (12) |
|
37 Mapping of Domains and Boundaries |
|
|
755 | (42) |
|
37.1 Unicity of Mappings and Bounds in the Unit Disk (Schwartz, 1890; Caratheodory, 1912; Borel, 1928) |
|
|
756 | (3) |
|
37.2 Existence of a Uniforly Convergent Subsequence (Vitali, 1903; Montel, 1910; Osgood, 1929) |
|
|
759 | (3) |
|
37.3 Simply Connected Region with at Least Two Boundary Points (Riemann, 1863) |
|
|
762 | (3) |
|
37.4 Mapping between Multiply Connected regions by Multivalent/Multivalued Functions |
|
|
765 | (5) |
|
37.5 Minimax, Reference, and Punctured Mappings |
|
|
770 | (2) |
|
37.6 Automorphism Group and Fundamental Regions |
|
|
772 | (3) |
|
37.7 Correspondence of Interiors and Boundaries for Compact and Noncompact Regions |
|
|
775 | (5) |
|
37.8 Interior and Exterior Integral Theorems (Cauchy, 1821; Schwartz, 1890) |
|
|
780 | (3) |
|
37.9 Harmonic Functions Defined by Boundary Values (Poisson, 1820; Dirichlet, 1850; robin, 1886; von Neumann, 1961) |
|
|
783 | (14) |
|
38 Hodograph for free jets |
|
|
797 | (56) |
|
38.1 Fields due to Potentials on Planes and Cylinders |
|
|
797 | (8) |
|
38.2 Width of the Vena Contracta of a Jet (Borda, 1766) |
|
|
805 | (2) |
|
38.3 Slit in a Wall and Reentrant Tube in a Reservoir (Helmholtz, 1868) |
|
|
807 | (5) |
|
38.4 Flat Plate Orthogonal to a Jet or to a Wall (Kirchhoff, 1869; Rayleigh, 1876a) |
|
|
812 | (5) |
|
38.5 Center of Pressure and Dividing Streamline on a Surfboard (Rayleigh, 1876b, 1891) |
|
|
817 | (7) |
|
38.6 Arrow or Bent Lamina in a Stream (Rethy, 1879; Bobyleff, 1881) |
|
|
824 | (6) |
|
38.7 Jet Attachment around a Wall (Coanda Effect) |
|
|
830 | (4) |
|
38.8 Fluidics: Deflection of a Jet by a Small Source |
|
|
834 | (5) |
|
38.9 Jets merging, Splitting, or Colliding with a Wall |
|
|
839 | (14) |
|
39 Essential Singularities, Roots, and periods |
|
|
853 | (44) |
|
39.1 Classification of Special and Singular Points |
|
|
853 | (3) |
|
39.2 Zeros, Poles, and Essential Singularities (Casorati, 1868; Weierstrass, 1876; Picard, 1880) |
|
|
856 | (4) |
|
39.3 Exceptional Value (Picard, 1879) and Infinite Number of Roots |
|
|
860 | (4) |
|
39.4 Triangular Coverings (Schwartz, 1890) and the Modular Function (Legendre) |
|
|
864 | (6) |
|
39.5 Dense Rays and Julia (1924) Theorem |
|
|
870 | (5) |
|
39.6 Cauchy Bounds and Landau (1904) Radius (Schottky, 1904; Caratheodory, 1912; Montel, 1927) |
|
|
875 | (7) |
|
39.7 Periods and Inversion of Hyperelliptic Integrals |
|
|
882 | (4) |
|
39.8 Logarithm, Exponential, and Circular/Hyperbolic Functions |
|
|
886 | (3) |
|
39.9 Elliptic Functions of Jacobi (1827) and Weierstrass (1895) |
|
|
889 | (8) |
|
40 Examples 40.1 to 40.20 |
|
|
897 | (56) |
| Bibliography |
|
953 | (10) |
| References |
|
963 | (6) |
| Index |
|
969 | |
