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Applied Complex Variables For Scientists And Engineers
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Preface page ix 1 Complex Numbers 1 1.1 Complex numbers and their representations 1 1.2 Algebraic properties of complex numbers 4 1.2.1 De Moivre’s theorem 7 1.3 Geometric properties of complex numbers 13 1.3.1 nth roots of unity 16 1.3.2 Symmetry with respect to a circle 17 1.4 Some topological definitions 23 1.5 Complex infinity and the Riemann sphere 29 1.5.1 The Riemann sphere and stereographic projection 30 1.6 Applications to electrical circuits 33 1.7 Problems 36 2 Analytic Functions 46 2.1 Functions of a complex variable 46 2.1.1 Velocity of fluid flow emanating from a source 48 2.1.2 Mapping properties of complex functions 50 2.1.3 Definitions of the exponential and trigonometric functions 53 2.2 Limit and continuity of complex functions 54 2.2.1 Limit of a complex function 54 2.2.2 Continuity of a complex function 58 2.3 Differentiation of complex functions 61 2.3.1 Complex velocity and acceleration 63 2.4 Cauchy–Riemann relations 64 2.4.1 Conjugate complex variables 69 v vi Contents 2.5 Analyticity 70 2.6 Harmonic functions 74 2.6.1 Harmonic conjugate 75 2.6.2 Steady state temperature distribution 80 2.6.3 Poisson’s equation 84 2.7 Problems 85 3 Exponential, Logarithmic and Trigonometric Functions 93 3.1 Exponential functions 93 3.1.1 Definition from the first principles 94 3.1.2 Mapping properties of the complex exponential function 97 3.2 Trigonometric and hyperbolic functions 97 3.2.1 Mapping properties of the complex sine function 102 3.3 Logarithmic functions 104 3.3.1 Heat source 106 3.3.2 Temperature distribution in the upper half-plane 108 3.4 Inverse trigonometric and hyperbolic functions 111 3.5 Generalized exponential, logarithmic, and power functions 115 3.6 Branch points, branch cuts and Riemann surfaces 118 3.6.1 Joukowski mapping 123 3.7 Problems 126 4 Complex Integration 133 4.1 Formulations of complex integration 133 4.1.1 Definite integral of a complex-valued function of a real variable 134 4.1.2 Complex integrals as line integrals 135 4.2 Cauchy integral theorem 142 4.3 Cauchy integral formula and its consequences 151 4.3.1 Derivatives of contour integrals 153 4.3.2 Morera’s theorem 157 4.3.3 Consequences of the Cauchy integral formula 158 4.4 Potential functions of conservative fields 162 4.4.1 Velocity potential and stream function of fluid flows 162 4.4.2 Electrostatic fields 175 4.4.3 Gravitational fields 179 4.5 Problems 183 Contents vii 5 Taylor and Laurent Series 194 5.1 Complex sequences and series 194 5.1.1 Convergence of complex sequences 194 5.1.2 Infinite series of complex numbers 196 5.1.3 Convergence tests of complex series 197 5.2 Sequences and series of complex functions 200 5.2.1 Convergence of series of complex functions 201 5.2.2 Power series 206 5.3 Taylor series 215 5.4 Laurent series 221 5.4.1 Potential flow past an obstacle 230 5.5 Analytic continuation 233 5.5.1 Reflection principle 236 5.6 Problems 238 6 Singularities and Calculus of Residues 248 6.1 Classification of singular points 248 6.2 Residues and the Residue Theorem 255 6.2.1 Computational formulas for evaluating residues 257 6.3 Evaluation of real integrals by residue calculus 268 6.3.1 Integrals of trigonometric functions over [0, 2π] 268 6.3.2 Integrals of rational functions 269 6.3.3 Integrals involving multi-valued functions 271 6.3.4 Miscellaneous types of integral 275 6.4 Fourier transforms 278 6.4.1 Fourier inversion formula 279 6.4.2 Evaluation of Fourier integrals 285 6.5 Cauchy principal value of an improper integral 288 6.6 Hydrodynamics in potential fluid flows 295 6.6.1 Blasius laws of hydrodynamic force and moment 295 6.6.2 Kutta–Joukowski’s lifting force theorem 299 6.7 Problems 300 7 Boundary Value Problems and Initial-Boundary Value Problems 311 7.1 Integral formulas of harmonic functions 312 7.1.1 Poisson integral formula 312 7.1.2 Schwarz integral formula 319 7.1.3 Neumann problems 324 7.2 The Laplace transform and its inversion 326 7.2.1 Bromwich integrals 330 viii Contents 7.3 Initial-boundary value problems 336 7.3.1 Heat conduction 337 7.3.2 Longitudinal oscillations of an elastic thin rod 341 7.4 Problems 346 8 Conformal Mappings and Applications 358 8.1 Conformal mappings 358 8.1.1 Invariance of the Laplace equation 364 8.1.2 Hodograph transformations 372 8.2 Bilinear transformations 375 8.2.1 Circle-preserving property 378 8.2.2 Symmetry-preserving property 381 8.2.3 Some special bilinear transformations 390 8.3 Schwarz–Christoffel transformations 399 8.4 Problems 409 Answers to Problems 419 Index 434 |
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