## Advanced engineering mathematics / Erwin Kreyszig.

Editor: New York : Wiley, c1993Edición: 7th edDescripción: 1 v. (varias paginaciones) : il. (algunas col.) ; 27 cmISBN: 0471553808 (hardcover)Tema(s): Mathematical physics | Engineering mathematicsOtra clasificación: 00A06Contents Part A. ORDINARY DIFFERENTIAL EQUATIONS [1] CHAPTER 1 First-Order Differential Equations [2] 1.1 Basic Concepts and Ideas, [2] 1.2 Separable Differential Equations, [10] 1.3 Modeling: Separable Equations, [13] 1.4 Reduction to Separable Form. Optional, [20] 1.5 Exact Differential Equations, [23] 1.6 Integrating Factors, [27] 1.7 Linear Differential Equations, [30] 1.8 Modeling: Electric Circuits, [37] 1.9 Orthogonal Trajectories of Curves. Optional, [43] 1.10 Approximate Solutions: Direction Fields, Iteration, [48] 1.11 Existence and Uniqueness of Solutions, [53] Chapter Review Questions and Problems, [58] Chapter Summary, [60] CHAPTER 2 * Second-Order Linear Differential Equations [62] 2.1 Homogeneous Linear Equations, [62] 2.2 Homogeneous Equations with Constant Coefficients, [69] 2.3 Case of Complex Roots. Complex Exponential Function, [73] 2.4 Differential Operators. Optional, [77] 2.5 Modeling: Free Oscillations (Mass-Spring System), [80] 2.6 Euler-Cauchy Equation, [90] 2.7 Existence and Uniqueness Theory. Wronskian, [93] 2.8 Nonhomogeneous Equations, [99] 2.9 Solution by Undetermined Coefficients, [102] 2.10 Solution by Variation of Parameters, [106] 2.11 Modeling: Forced Oscillations. Resonance, [109] 2.12 Modeling of Electric Circuits, [116] 2.13 Complex Method for Particular Solutions. Optional, [121] Chapter Review Questions and Problems, [124] Chapter Summary, [126] CHAPTER 3 Higher Order Linear Differential Equations [128] 3.1 Homogeneous Linear Equations, [128] 3.2 Homogeneous Equations with Constant Coefficients, [136] 3.3 Nonhomogeneous Equations, [141] 3.4 Method of Undetermined Coefficients, [143] 3.5 Method of Variation of Parameters, [146] Chapter Review Questions and Problems, [150] Chapter Summary, [151] CHAPTER 4 Systems of Differential Equations. Phase Plane, Stability [152] 4.0 Introduction: Vectors, Matrices, [152] 4.1 Introductory Examples, [158] 4.2 Basic Concepts and Theory, [163] 4.3 Homogeneous Linear Systems with Constant Coefficients, [166] 4.4 Phase Plane, Critical Points, Stability, [176] 4.5 Phase Plane Methods for Nonlinear Systems, [180] 4.6 Nonhomogeneous Linear Systems, [186] Chapter Review Questions and Problems, [193] Chapter Summary, [195] CHAPTER 5 Series Solutions of Differential Equations. Special Functions [197] 5.1 Power Series Method, [197] 5.2 Theory of the Power Series Method, [201] 5.3 Legendre’s Equation. Legendre Polynomials Pn(x), [209] 5.4 Frobenius Method, [215] 5.5 Bessel’s Equation. Bessel Functions Jv(x), [225] 5.6 Further Properties of Jv(x), [232] 5.7 Bessel Functions of the Second Kind, [236] 5.8 Sturm-Liouville Problems. Orthogonality, [241] 5.9 Eigenfunction Expansions, [249] Chapter Review Questions and Problems, [258] Chapter Summary, [259] CHAPTER 6 Laplace Transforms [261] 6.1 Laplace Transform. Inverse Transform. Linearity, [262] 6.2 Transforms of Derivatives and Integrals, [268] 6.3 s-Shifting, t-Shifting, Unit Step Function, [275] 6.4 Further Applications. Dirac’s Delta Function, [284] 6.5 Differentiation and Integration of Transforms, [289] 6.6 Convolution. Integral Equations, [293] 6.7 Partial Fractions. Systems of Differential Equations, [299] 6.8 Periodic Functions. Further Applications, [309] 6.9 Laplace Transform: General Formulas, [317] 6.10 Table of Laplace Transforms, [318] Chapter Review Questions and Problems, [320] Chapter Summary, [323] Part B. LINEAR ALGEBRA, VECTOR CALCULUS [325] CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants [326] 7.1 Basic Concepts, [327] 7.2 Matrix Addition, Scalar Multiplication, [330] 7.3 Matrix Multiplication, [334] 7.4 Linear Systems of Equations. Gauss Elimination, [344] 7.5 Linear Independence. Vector Space. Rank of a Matrix, [354] 7.6 Linear Systems: General Properties of Solutions, [361] 7.7 Inverse of a Matrix, [365] 7.8 Determinants, [370] 7.9 Rank in Terms of Determinants. Cramer’s Rule, [380] 7.10 Eigenvalues, Eigenvectors, [386] 7.11 Some Applications of Eigenvalue Problems, [392] 7.12 Symmetric, Skew-Symmetric, and Orthogonal Matrices, [397] 7.13 Hermitian, Skew-Hermitian, and Unitary Matrices, [401] 7.14 Properties of Eigenvectors. Diagonalization, [408] 7.15 Vector Spaces, Inner Product Spaces, Linear Transformations. Optional, [414] Chapter Review Questions and Problems, [422] Chapter Summary, [426] CHAPTER 8 Vector Differential Calculus. Grad, Div, Curl [428] 8.1 Vector Algebra in 2-Space and 3-Space, [429] 8.2 Inner Product (Dot Product), [436] 8.3 Vector Product (Cross Product), [442] 8.4 Vector and Scalar Functions and Fields. Derivatives, [451] 8.5 Curves. Tangents. Arc Length, [457] 8.6 Velocity and Acceleration, [464] 8.7 Curvature and Torsion of a Curve. Optional, [469] 8.8 Review from Calculus in Several Variables. Optional, [472] 8.9 Gradient of a Scalar Field. Directional Derivative, [475] 8.10 Divergence of a Vector Field, [482] 8.11 Curl of a Vector Field, [486] 8.12 Grad, Div, Curl in Curvilinear Coordinates. Optional, [488] Chapter Review Questions and Problems, [494] Chapter Summary, [497] CHAPTER 9 Vector Integral Calculus. Integral Theorems [500] 9.1 Line Integrals, [500] 9.2 Line Integrals Independent of Path, [507] 9.3 From Calculus: Double Integrals. Optional, [515] 9.4 Green’s Theorem in the Plane, [522] 9.5 Surfaces for Surface Integrals, [529] 9.6 Surface Integrals, [534] 9.7 Triple Integrals. Divergence Theorem of Gauss, [544] 9.8 Further Applications of the Divergence Theorem, [550] 9.9 Stokes’s Theorem, [556] Chapter Review Questions and Problems, [562] Chapter Summary, [563] Part C. FOURIER ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS [565] CHAPTER 10 Fourier Series, Integrals, and Transforms [566] 10.1 Periodic Functions. Trigonometric Series, [566] 10.2 Fourier Series, [569] 10.3 Functions of Any Period p = 2L, [577] 10.4 Even and Odd Functions, [580] 10.5 Half-Range Expansions, [585] 10.6 Complex Fourier Series. Optional, [588] 10.7 Forced Oscillations, [591] 10.8 Approximation by Trigonometric Polynomials, [595] 10.9 Fourier Integrals, [598] 10.10 Fourier Cosine and Sine Transforms, [606] 10.11 Fourier Transform, [611] 10.12 Tables of Transforms, [619] Chapter Review Questions and Problems, [622] Chapter Summary, [624] CHAPTER 11 Partial Differential Equations [626] 11.1 Basic Concepts, [627] 11.2 Modeling: Vibrating String, Wave Equation, [629] 11.3 Separation of Variables. Use of Fourier Series, [631] 11.4 D’Alembert’s Solution of the Wave Equation, [639] 11.5 Heat Equation: Solution by Fourier Series, [645] 11.6 Heat Equation: Solution by Fourier Integrals, [657] 11.7 Modeling: Membrane, Two-Dimensional Wave Equation, [662] 11.8 Rectangular Membrane, Use of Double Fourier Series, [664] 11.9 Laplacian in Polar Coordinates, [672] 11.10 Circular Membrane. Use of Fourier-Bessel Series, [675] 11.11 Laplace’s Equation. Potential, [682] 11.12 Laplacian in Spherical Coordinates. Legendre’s Equation, [686] 11.13 Solution by Laplace Transforms, [692] 11.14 Solution by Fourier Transforms, [696] Chapter Review Questions and Problems, [700] Chapter Summary, [702]

Part D. COMPLEX ANALYSIS [705] CHAPTER 12 Complex Numbers. Complex Analytic Functions [706] 12.1 Complex Numbers. Complex Plane, [706] 12.2 Polar Form of Complex Numbers. Powers and Roots, [711] 12.3 Curves and Regions in the Complex Plane, [718] 12.4 Limit. Derivative. Analytic Function, [721] 12.5 Cauchy-Riemann Equations, [726] 12.6 Exponential Function, [731] 12.7 Trigonometric Functions, Hyperbolic Functions, [735] 12.8 Logarithm. General Power, [739] 12.9 Mapping by Special Functions. Optional, [743] Chapter Review Questions and Problems, [747] Chapter Summary, [749] CHAPTER 13 Complex Integration [751] 13.1 Line Integral in the Complex Plane, [751] 13.2 Two Integration Methods. Examples, [755] 13.3 Cauchy’s Integral Theorem, [761] 13.4 Existence of Indefinite Integral, [768] 13.5 Cauchy’s Integral Formula, [770] 13.6 Derivatives of Analytic Functions, [774] Chapter Review Questions and Problems, [778] Chapter Summary, [780] CHAPTER 14 Power Series, Taylor Series, Laurent Series [781] 14.1 Sequences, Series, Convergence Tests, [781] 14.2 Power Series, [791] 14.3 Functions Given by Power Series, [796] 14.4 Taylor Series, [802] 14.5 Power Series: Practical Methods, [808] 14.6 Uniform Convergence, [812] 14.7 Laurent Series, [821] 14.8 Singularities and Zeros. Infinity, [828] Chapter Review Questions and Problems, [834] Chapter Summary, [835] CHAPTER 15 Residue Integration Method [837] 15.1 Residues, [837] 15.2 Residue Theorem, [842] 15.3 Evaluation of Real Integrals, [845] 15.4 Further Types of Real Integrals, [849] Chapter Review Questions and Problems, [856] Chapter Summary, [857] CHAPTER 16 Conformal Mapping [859] 16.1 Conformal Mapping, [859] 16.2 Linear Fractional Transformations, [863] 16.3 Special Linear Fractional Transformations, [868] 16.4 Mapping by Other Functions, [873] 16.5 Riemann Surfaces, [878] Chapter Review Questions and Problems, [883] Chapter Summary, [884] CHAPTER 17 Complex Analysis Applied to Potential Theory [885] 17.1 Electrostatic Fields, [886] 17.2 Use of Conformal Mapping, [890] 17.3 Heat Problems, [894] 17.4 Fluid Flow, [898] 17.5 Poisson’s Integral Formula, [905] 17.6 General Properties of Harmonic Functions, [910] Chapter Review Questions and Problems, [914] Chapter Summary, [915] Part E. NUMERICAL METHODS [916] CHAPTER 18 Numerical Methods in General [918] 18.1 Introduction, [919] 18.2 Solution of Equations by Iteration, [925] 18.3 Interpolation, [936] 18.4 Splines, [949] 18.5 Numerical Integration and Differentiation, [957] Chapter Review Questions and Problems, [968] Chapter Summary, [970] CHAPTER 19 Numerical Methods in Linear Algebra [972] 19.1 Linear Systems: Gauss Elimination, [972] 19.2 Linear Systems: LU-Factorization, Matrix Inversion, [981] 19.3 Linear Systems: Solution by Iteration, [986] 19.4 Linear Systems: Ill-Conditioning, Norms, [993] 19.5 Method of Least Squares, [1000] 19.6 Matrix Eigenvalue Problems: Introduction, [1004] 19.7 Inclusion of Matrix Eigenvalues, [1006] 19.8 Eigenvalues by Iteration (Power Method), [1012] 19.9 Deflation of a Matrix, [1016] 19.10 Householder Tridiagonalization and QR-Factorization, [1019] Chapter Review Questions and Problems, [1029] Chapter Summary, [1031] CHAPTER 20 Numerical Methods for Differential Equations [1034] 20.1 Methods for First-Order Differential Equations, [1034] 20.2 Multistep Methods, [1044] 20.3 Methods for Second-Order Differential Equations, [1048] 20.4 Numerical Methods for Elliptic Partial Differential Equations, [1055] 20.5 Neumann and Mixed Problems. Irregular Boundary, [1065] 20.6 Methods for Parabolic Equations, [1070] 20.7 Methods for Hyperbolic Equations, [1075] Chapter Review Questions and Problems, [1078] Chapter Summary, [1081] Part F. OPTIMIZATION, GRAPHS [1083] CHAPTER 21 Unconstrained Optimization, Linear Programming [1084] 21.1 Basic Concepts. Unconstrained Optimization, [1084] 21.2 Linear Programming, [1088] 21.3 Simplex Method, [1092] 21.4 Simplex Method: Degeneracy, Difficulties in Starting, [1097] Chapter Review Questions and Problems, [1102] Chapter Summary, [1104] CHAPTER 22 Graphs and Combinatorial Optimization [1105] 22.1 Graphs and Digraphs, [1105] 22.2 Shortest Path Problems. Complexity, [1110] 22.3 Bellman’s Optimality Principle. Dijkstra’s Algorithm, [1116] 22.4 Shortest Spanning Trees. Kruskal’s Greedy Algorithm, [1120] 22.5 Prim’s Algorithm for Shortest Spanning Trees, [1125] 22.6 Networks. Flow Augmenting Paths, [1128] 22.7 Ford-Fulkerson Algorithm for Maximum Flow, [1134] 22.8 Assignment Problems. Bipartite Matching, [1138] Chapter Review Questions and Problems, [1145] Chapter Summary, [1147] PartG. PROBABILITY AND STATISTICS [1148] CHAPTER 23 Probability Theory [1149] 23.1 Experiments, Outcomes, Events, [1149] 23.2 Probability, [1153] 23.3 Permutations and Combinations, [1160] 23.4 Random Variables, Probability Distributions, [1165] 23.5 Mean and Variance of a Distribution, [1173] 23.6 Binomial, Poisson, and Hypergeometric Distributions, [1178] 23.7 Normal Distribution, [1184] 23.8 Distributions of Several Random Variables, [1191] Chapter Review Questions and Problems, [1200] Chapter Summary, [1203] CHAPTER 24 Mathematical Statistics [1205] 24.1 Nature and Purpose of Statistics, [1205] 24.2 Random Sampling. Random Numbers, [1207] 24.3 Processing of Samples, [1209] 24.4 Sample Mean, Sample Variance, [1216] 24.5 Estimation of Parameters, [1219] 24.6 Confidence Intervals, [1222] 24.7 Testing of Hypotheses, Decisions, [1232] 24.8 Quality Control, [1244] 24.9 Acceptance Sampling, [1249] 24.10 Goodness of Fit. x2-Test, [1255] 24.11 Nonparametric Tests, [1258] 24.12 Pairs of Measurements. Fitting Straight Lines, [1261] Chapter Review Questions and Problems, [1267] Chapter Summary, [1270] APPENDIX 1 References A1 APPENDIX 2 Answers to Odd-Numbered Problems A6 APPENDIX 3 Auxiliary Material A66 A3.1 Formulas for Special Functions, A66 A3.2 Partial Derivatives, A72 A3.3 Sequences and Series, A74 APPENDIX 4 Additional Proofs A77 APPENDIX 5 Tables A97 INDEX I1

Item type | Home library | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
---|---|---|---|---|---|---|---|

Libros | Instituto de Matemática, CONICET-UNS | 00A06 K92-7 (Browse shelf) | Available | A-7047 |

Incluye referencias bibliográficas e índice.

Contents --

Part A. ORDINARY DIFFERENTIAL EQUATIONS [1] --

CHAPTER 1 First-Order Differential Equations [2] --

1.1 Basic Concepts and Ideas, [2] --

1.2 Separable Differential Equations, [10] --

1.3 Modeling: Separable Equations, [13] --

1.4 Reduction to Separable Form. Optional, [20] --

1.5 Exact Differential Equations, [23] --

1.6 Integrating Factors, [27] --

1.7 Linear Differential Equations, [30] --

1.8 Modeling: Electric Circuits, [37] --

1.9 Orthogonal Trajectories of Curves. Optional, [43] --

1.10 Approximate Solutions: Direction Fields, Iteration, [48] --

1.11 Existence and Uniqueness of Solutions, [53] --

Chapter Review Questions and Problems, [58] --

Chapter Summary, [60] --

CHAPTER 2 * Second-Order Linear Differential Equations [62] --

2.1 Homogeneous Linear Equations, [62] --

2.2 Homogeneous Equations with Constant Coefficients, [69] --

2.3 Case of Complex Roots. Complex Exponential Function, [73] --

2.4 Differential Operators. Optional, [77] --

2.5 Modeling: Free Oscillations (Mass-Spring System), [80] --

2.6 Euler-Cauchy Equation, [90] --

2.7 Existence and Uniqueness Theory. Wronskian, [93] --

2.8 Nonhomogeneous Equations, [99] --

2.9 Solution by Undetermined Coefficients, [102] --

2.10 Solution by Variation of Parameters, [106] --

2.11 Modeling: Forced Oscillations. Resonance, [109] --

2.12 Modeling of Electric Circuits, [116] --

2.13 Complex Method for Particular Solutions. Optional, [121] --

Chapter Review Questions and Problems, [124] --

Chapter Summary, [126] --

CHAPTER 3 Higher Order Linear Differential Equations [128] --

3.1 Homogeneous Linear Equations, [128] --

3.2 Homogeneous Equations with Constant Coefficients, [136] --

3.3 Nonhomogeneous Equations, [141] --

3.4 Method of Undetermined Coefficients, [143] --

3.5 Method of Variation of Parameters, [146] --

Chapter Review Questions and Problems, [150] --

Chapter Summary, [151] --

CHAPTER 4 Systems of Differential Equations. --

Phase Plane, Stability [152] --

4.0 Introduction: Vectors, Matrices, [152] --

4.1 Introductory Examples, [158] --

4.2 Basic Concepts and Theory, [163] --

4.3 Homogeneous Linear Systems with Constant Coefficients, [166] --

4.4 Phase Plane, Critical Points, Stability, [176] --

4.5 Phase Plane Methods for Nonlinear Systems, [180] --

4.6 Nonhomogeneous Linear Systems, [186] --

Chapter Review Questions and Problems, [193] --

Chapter Summary, [195] --

CHAPTER 5 Series Solutions of Differential Equations. --

Special Functions [197] --

5.1 Power Series Method, [197] --

5.2 Theory of the Power Series Method, [201] --

5.3 Legendre’s Equation. Legendre Polynomials Pn(x), [209] --

5.4 Frobenius Method, [215] --

5.5 Bessel’s Equation. Bessel Functions Jv(x), [225] --

5.6 Further Properties of Jv(x), [232] --

5.7 Bessel Functions of the Second Kind, [236] --

5.8 Sturm-Liouville Problems. Orthogonality, [241] --

5.9 Eigenfunction Expansions, [249] --

Chapter Review Questions and Problems, [258] --

Chapter Summary, [259] --

CHAPTER 6 Laplace Transforms [261] --

6.1 Laplace Transform. Inverse Transform. Linearity, [262] --

6.2 Transforms of Derivatives and Integrals, [268] --

6.3 s-Shifting, t-Shifting, Unit Step Function, [275] --

6.4 Further Applications. Dirac’s Delta Function, [284] --

6.5 Differentiation and Integration of Transforms, [289] --

6.6 Convolution. Integral Equations, [293] --

6.7 Partial Fractions. Systems of Differential Equations, [299] --

6.8 Periodic Functions. Further Applications, [309] --

6.9 Laplace Transform: General Formulas, [317] --

6.10 Table of Laplace Transforms, [318] --

Chapter Review Questions and Problems, [320] --

Chapter Summary, [323] --

Part B. LINEAR ALGEBRA, VECTOR CALCULUS [325] --

CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants [326] --

7.1 Basic Concepts, [327] --

7.2 Matrix Addition, Scalar Multiplication, [330] --

7.3 Matrix Multiplication, [334] --

7.4 Linear Systems of Equations. Gauss Elimination, [344] --

7.5 Linear Independence. Vector Space. Rank of a Matrix, [354] --

7.6 Linear Systems: General Properties of Solutions, [361] --

7.7 Inverse of a Matrix, [365] --

7.8 Determinants, [370] --

7.9 Rank in Terms of Determinants. Cramer’s Rule, [380] --

7.10 Eigenvalues, Eigenvectors, [386] --

7.11 Some Applications of Eigenvalue Problems, [392] --

7.12 Symmetric, Skew-Symmetric, and Orthogonal Matrices, [397] --

7.13 Hermitian, Skew-Hermitian, and Unitary Matrices, [401] --

7.14 Properties of Eigenvectors. Diagonalization, [408] --

7.15 Vector Spaces, Inner Product Spaces, Linear Transformations. Optional, [414] --

Chapter Review Questions and Problems, [422] --

Chapter Summary, [426] --

CHAPTER 8 Vector Differential Calculus. Grad, Div, Curl [428] --

8.1 Vector Algebra in 2-Space and 3-Space, [429] --

8.2 Inner Product (Dot Product), [436] --

8.3 Vector Product (Cross Product), [442] --

8.4 Vector and Scalar Functions and Fields. Derivatives, [451] --

8.5 Curves. Tangents. Arc Length, [457] --

8.6 Velocity and Acceleration, [464] --

8.7 Curvature and Torsion of a Curve. Optional, [469] --

8.8 Review from Calculus in Several Variables. Optional, [472] --

8.9 Gradient of a Scalar Field. Directional Derivative, [475] --

8.10 Divergence of a Vector Field, [482] --

8.11 Curl of a Vector Field, [486] --

8.12 Grad, Div, Curl in Curvilinear Coordinates. Optional, [488] --

Chapter Review Questions and Problems, [494] --

Chapter Summary, [497] --

CHAPTER 9 Vector Integral Calculus. Integral Theorems [500] --

9.1 Line Integrals, [500] --

9.2 Line Integrals Independent of Path, [507] --

9.3 From Calculus: Double Integrals. Optional, [515] --

9.4 Green’s Theorem in the Plane, [522] --

9.5 Surfaces for Surface Integrals, [529] --

9.6 Surface Integrals, [534] --

9.7 Triple Integrals. Divergence Theorem of Gauss, [544] --

9.8 Further Applications of the Divergence Theorem, [550] --

9.9 Stokes’s Theorem, [556] --

Chapter Review Questions and Problems, [562] --

Chapter Summary, [563] --

Part C. FOURIER ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS [565] --

CHAPTER 10 Fourier Series, Integrals, and Transforms [566] --

10.1 Periodic Functions. Trigonometric Series, [566] --

10.2 Fourier Series, [569] --

10.3 Functions of Any Period p = 2L, [577] --

10.4 Even and Odd Functions, [580] --

10.5 Half-Range Expansions, [585] --

10.6 Complex Fourier Series. Optional, [588] --

10.7 Forced Oscillations, [591] --

10.8 Approximation by Trigonometric Polynomials, [595] --

10.9 Fourier Integrals, [598] --

10.10 Fourier Cosine and Sine Transforms, [606] --

10.11 Fourier Transform, [611] --

10.12 Tables of Transforms, [619] --

Chapter Review Questions and Problems, [622] --

Chapter Summary, [624] --

CHAPTER 11 Partial Differential Equations [626] --

11.1 Basic Concepts, [627] --

11.2 Modeling: Vibrating String, Wave Equation, [629] --

11.3 Separation of Variables. Use of Fourier Series, [631] --

11.4 D’Alembert’s Solution of the Wave Equation, [639] --

11.5 Heat Equation: Solution by Fourier Series, [645] --

11.6 Heat Equation: Solution by Fourier Integrals, [657] --

11.7 Modeling: Membrane, Two-Dimensional Wave Equation, [662] --

11.8 Rectangular Membrane, Use of Double Fourier Series, [664] --

11.9 Laplacian in Polar Coordinates, [672] --

11.10 Circular Membrane. Use of Fourier-Bessel Series, [675] --

11.11 Laplace’s Equation. Potential, [682] --

11.12 Laplacian in Spherical Coordinates. Legendre’s Equation, [686] --

11.13 Solution by Laplace Transforms, [692] --

11.14 Solution by Fourier Transforms, [696] --

Chapter Review Questions and Problems, [700] --

Chapter Summary, [702] --

Part D. COMPLEX ANALYSIS [705] --

CHAPTER 12 Complex Numbers. Complex Analytic Functions [706] --

12.1 Complex Numbers. Complex Plane, [706] --

12.2 Polar Form of Complex Numbers. Powers and Roots, [711] --

12.3 Curves and Regions in the Complex Plane, [718] --

12.4 Limit. Derivative. Analytic Function, [721] --

12.5 Cauchy-Riemann Equations, [726] --

12.6 Exponential Function, [731] --

12.7 Trigonometric Functions, Hyperbolic Functions, [735] --

12.8 Logarithm. General Power, [739] --

12.9 Mapping by Special Functions. Optional, [743] --

Chapter Review Questions and Problems, [747] --

Chapter Summary, [749] --

CHAPTER 13 Complex Integration [751] --

13.1 Line Integral in the Complex Plane, [751] --

13.2 Two Integration Methods. Examples, [755] --

13.3 Cauchy’s Integral Theorem, [761] --

13.4 Existence of Indefinite Integral, [768] --

13.5 Cauchy’s Integral Formula, [770] --

13.6 Derivatives of Analytic Functions, [774] --

Chapter Review Questions and Problems, [778] --

Chapter Summary, [780] --

CHAPTER 14 Power Series, Taylor Series, Laurent Series [781] --

14.1 Sequences, Series, Convergence Tests, [781] --

14.2 Power Series, [791] --

14.3 Functions Given by Power Series, [796] --

14.4 Taylor Series, [802] --

14.5 Power Series: Practical Methods, [808] --

14.6 Uniform Convergence, [812] --

14.7 Laurent Series, [821] --

14.8 Singularities and Zeros. Infinity, [828] --

Chapter Review Questions and Problems, [834] --

Chapter Summary, [835] --

CHAPTER 15 Residue Integration Method [837] --

15.1 Residues, [837] --

15.2 Residue Theorem, [842] --

15.3 Evaluation of Real Integrals, [845] --

15.4 Further Types of Real Integrals, [849] --

Chapter Review Questions and Problems, [856] --

Chapter Summary, [857] --

CHAPTER 16 Conformal Mapping [859] --

16.1 Conformal Mapping, [859] --

16.2 Linear Fractional Transformations, [863] --

16.3 Special Linear Fractional Transformations, [868] --

16.4 Mapping by Other Functions, [873] --

16.5 Riemann Surfaces, [878] --

Chapter Review Questions and Problems, [883] --

Chapter Summary, [884] --

CHAPTER 17 Complex Analysis Applied to Potential Theory [885] --

17.1 Electrostatic Fields, [886] --

17.2 Use of Conformal Mapping, [890] --

17.3 Heat Problems, [894] --

17.4 Fluid Flow, [898] --

17.5 Poisson’s Integral Formula, [905] --

17.6 General Properties of Harmonic Functions, [910] --

Chapter Review Questions and Problems, [914] --

Chapter Summary, [915] --

Part E. NUMERICAL METHODS [916] --

CHAPTER 18 Numerical Methods in General [918] --

18.1 Introduction, [919] --

18.2 Solution of Equations by Iteration, [925] --

18.3 Interpolation, [936] --

18.4 Splines, [949] --

18.5 Numerical Integration and Differentiation, [957] --

Chapter Review Questions and Problems, [968] --

Chapter Summary, [970] --

CHAPTER 19 Numerical Methods in Linear Algebra [972] --

19.1 Linear Systems: Gauss Elimination, [972] --

19.2 Linear Systems: LU-Factorization, Matrix Inversion, [981] --

19.3 Linear Systems: Solution by Iteration, [986] --

19.4 Linear Systems: Ill-Conditioning, Norms, [993] --

19.5 Method of Least Squares, [1000] --

19.6 Matrix Eigenvalue Problems: Introduction, [1004] --

19.7 Inclusion of Matrix Eigenvalues, [1006] --

19.8 Eigenvalues by Iteration (Power Method), [1012] --

19.9 Deflation of a Matrix, [1016] --

19.10 Householder Tridiagonalization and QR-Factorization, [1019] --

Chapter Review Questions and Problems, [1029] --

Chapter Summary, [1031] --

CHAPTER 20 Numerical Methods for Differential Equations [1034] --

20.1 Methods for First-Order Differential Equations, [1034] --

20.2 Multistep Methods, [1044] --

20.3 Methods for Second-Order Differential Equations, [1048] --

20.4 Numerical Methods for Elliptic Partial Differential Equations, [1055] --

20.5 Neumann and Mixed Problems. Irregular Boundary, [1065] --

20.6 Methods for Parabolic Equations, [1070] --

20.7 Methods for Hyperbolic Equations, [1075] --

Chapter Review Questions and Problems, [1078] --

Chapter Summary, [1081] --

Part F. OPTIMIZATION, GRAPHS [1083] --

CHAPTER 21 Unconstrained Optimization, Linear Programming [1084] --

21.1 Basic Concepts. Unconstrained Optimization, [1084] --

21.2 Linear Programming, [1088] --

21.3 Simplex Method, [1092] --

21.4 Simplex Method: Degeneracy, Difficulties in Starting, [1097] --

Chapter Review Questions and Problems, [1102] --

Chapter Summary, [1104] --

CHAPTER 22 Graphs and Combinatorial Optimization [1105] --

22.1 Graphs and Digraphs, [1105] --

22.2 Shortest Path Problems. Complexity, [1110] --

22.3 Bellman’s Optimality Principle. Dijkstra’s Algorithm, [1116] --

22.4 Shortest Spanning Trees. Kruskal’s Greedy Algorithm, [1120] --

22.5 Prim’s Algorithm for Shortest Spanning Trees, [1125] --

22.6 Networks. Flow Augmenting Paths, [1128] --

22.7 Ford-Fulkerson Algorithm for Maximum Flow, [1134] --

22.8 Assignment Problems. Bipartite Matching, [1138] --

Chapter Review Questions and Problems, [1145] --

Chapter Summary, [1147] --

PartG. PROBABILITY AND STATISTICS [1148] --

CHAPTER 23 Probability Theory [1149] --

23.1 Experiments, Outcomes, Events, [1149] --

23.2 Probability, [1153] --

23.3 Permutations and Combinations, [1160] --

23.4 Random Variables, Probability Distributions, [1165] --

23.5 Mean and Variance of a Distribution, [1173] --

23.6 Binomial, Poisson, and Hypergeometric Distributions, [1178] --

23.7 Normal Distribution, [1184] --

23.8 Distributions of Several Random Variables, [1191] --

Chapter Review Questions and Problems, [1200] --

Chapter Summary, [1203] --

CHAPTER 24 Mathematical Statistics [1205] --

24.1 Nature and Purpose of Statistics, [1205] --

24.2 Random Sampling. Random Numbers, [1207] --

24.3 Processing of Samples, [1209] --

24.4 Sample Mean, Sample Variance, [1216] --

24.5 Estimation of Parameters, [1219] --

24.6 Confidence Intervals, [1222] --

24.7 Testing of Hypotheses, Decisions, [1232] --

24.8 Quality Control, [1244] --

24.9 Acceptance Sampling, [1249] --

24.10 Goodness of Fit. x2-Test, [1255] --

24.11 Nonparametric Tests, [1258] --

24.12 Pairs of Measurements. Fitting Straight Lines, [1261] --

Chapter Review Questions and Problems, [1267] --

Chapter Summary, [1270] --

APPENDIX 1 References A1 --

APPENDIX 2 Answers to Odd-Numbered Problems A6 --

APPENDIX 3 Auxiliary Material A66 --

A3.1 Formulas for Special Functions, A66 --

A3.2 Partial Derivatives, A72 --

A3.3 Sequences and Series, A74 --

APPENDIX 4 Additional Proofs A77 --

APPENDIX 5 Tables A97 --

INDEX I1 --

MR, MR1190925

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