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## Advanced engineering mathematics / Erwin Kreyszig.

Editor: New York : Wiley, c1993Edición: 7th edDescripción: 1 v. (varias paginaciones) : il. (algunas col.) ; 27 cmISBN: 0471553808 (hardcover)Otra clasificación: 00A06
Contenidos:
``` 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```
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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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