Handbook of Applied Mathematics: Selected Results and Methods.

1 Formulas from Algebra, Trigonometry and Analytic Geometry.- 1.1 The Real Number System.- 1.2 The Complex Number System.- 1.3 Inequalities.- 1.4 Powers and Logarithms.- 1.5 Polynomial Equations.- 1.6 Rational Functions and Partial Fractions.- 1.7 Determinants and Solution of Systems of Linear Equations.- 1.8 Progressions.- 1.9 Binomial Theorem, Permutations and Combinations.- 1.10 The Trigonometric Functions.- 1.11 Analytic Geometry of Two-Space.- 1.12 Analytic Geometry of Three-Space.- 1.13 References and Bibliography.- 2 Elements of Analysis.- 2.1 Sequences.- 2.2 Infinite Series.- 2.3 Functions, Limits, Continuity.- 2.4 The Derivative.- 2.5 The Definite Integral.- 2.6 Methods of Integration.- 2.7 Improper Integrals.- 2.8 Partial Differentiation.- 2.9 Multiple Integrals.- 2.10 Infinite Products.- 2.11 Fourier Series.- 2.12 References and Bibliography.- 3 Vector Analysis.- 3.0 Introduction.- 3.1 Coordinate Systems.- 3.2 Vector Algebra.- 3.3 Vector Calculus.- 3.4 Successive Operations.- 3.5 Vector Fields.- 3.6 Summary.- 3.7 Bibliography.- 4 Tensors.- 4.0 Introduction.- 4.1 Vectors in Euclidean 3-D.- 4.2 Tensors in Euclidean 3-D.- 4.3 General Curvilinear Coordinates in Euclidean 3-D.- 4.4 Tensor Calculus.- 4.5 Theory of Surfaces.- 4.6 Classical Interlude.- 4.7 An Application: Continuum Mechanics.- 4.8 Tensors in n-Space.- 4.9 Bibliography.- 5 Functions of a Complex Variable.- 5.0 Introduction.- 5.1 Preliminaries.- 5.2 Analytic Functions.- 5.3 Singularities and Expansions.- 5.4 Residues and Contour Integrals.- 5.5 Harmonic Functions and Conformal Mapping.- 5.6 Acknowledgments.- 5.7 References and Bibliography.- 6 Ordinary Differential and Difference Equations.- 6.0 Introduction.- 6.1 Basic Concepts.- 6.2 First-Order Linear Differential Equations.- 6.3 Second Order Linear Differential Equations with Constant Coefficients.- 6.4 Second Order Linear Differential Equations with Variable Coefficients.- 6.5 Linear Equations of High Order and Systems of Equations.- 6.6 Eigenvalue Problems.- 6.7 Nonlinear Ordinary Differential Equations.- 6.8 Approximate Methods.- 6.9 Ordinary Difference Equations.- 6.10 References.- 7 Special Functions.- 7.0 Introduction.- 7.1 Exponential Integral and Related Functions.- 7.2 Gamma Function and Related Functions.- 7.3 Error Function and Related Functions.- 7.4 Bessel Functions.- 7.5 Modified Bessel Functions.- 7.6 Orthogonal Polynomials.- 7.7 Hypergeometric Functions and Legendre Functions.- 7.8 Elliptic Integrals and Functions.- 7.9 Other Special Functions.- 7.10 References and Bibliography.- 8 First Order Partial Differential Equations.- 8.0 Introduction.- 8.1 Examples of First Order Partial Differential Equations.- 8.2 Geometrical Concepts, Qualitative Results.- 8.3 Quasilinear Equations.- 8.4 Nonlinear Equations.- 8.5 References.- 9 Partial Differential Equations of Second and Higher Order.- 9.0 Survey of Contents.- 9.1 Derivation Examples.- 9.2 The Second-Order Linear Equation in Two Independent Variables.- 9.3 More General Equations.- 9.4 Series Solutions.- 9.5 Transform Methods.- 9.6 The Perturbation Idea.- 9.7 Change of Variable.- 9.8 Green's Function.- 9.9 Potential Theory.- 9.10 Eigenvalue Problems.- 9.11 Characteristics.- 9.12 Variational Methods.- 9.13 Numerical Techniques.- 9.14 References.- 10 Integral Equations.- 10.1 Introduction.- 10.2 Definitions and Classifications.- 10.3 Origin of Integral Equations.- 10.4 Nonsingular Linear Integral Equations.- 10.5 Singular Linear Integral Equations.- 10.6 Approximate Solution of Integral Equations.- 10.7 Nonlinear Integral Equations.- 10.8 References.- 11 Transform Methods.- 11.0 Introduction.- 11.1 Fourier's Integral Formula.- 11.2 Laplace Transforms.- 11.3 Linearity, Superposition, Representation Formulas.- 11.4 The Wiener-Hopf Technique.- 11.5 Abel's Integral Equation, Fractional Integrals, Weyl Transforms.- 11.6 Poisson's Formula, Summation of Series.- 11.7 Hilbert Transforms, Riemann-Hilbert Problem.- 11.8 Finite Transforms.- 11.9 Asymptotic Results.- 11.10 Operational Formulas.- 11.11 References.- 12 Asymptotic Methods.- 12.1 Definitions.- 12.2 Integrals of a Real Variable.- 12.3 Contour Integrals.- 12.4 Further Methods for Integrals.- 12.5 Sums and Sequences.- 12.6 The Liouville-Green (or JWKB) Approximation.- 12.7 Differential Equations with Irregular Singularities.- 12.8 Differential Equations with a Parameter.- 12.9 Estimation of Remainder Terms.- 12.10 References and Bibliography.- 13 Oscillations.- 13.0 Introduction.- 13.1 Lagrange Equations.- 13.2 Conservative Linear Systems, Direct Coupled.- 13.3 Systems with Gyroscopic Coupling.- 13.4 Mathieu-Hill Systems.- 13.5 Oscillations with Weak Nonlinearities.- 13.6 Oscillators Coupled by Weak Nonlinearity.- 13.7 References and Bibliography.- 14 Perturbation Methods.- 14.1 Introduction.- 14.2 Perturbation Methods for Ordinary Differential Equations.- 14.3 Partial Differential Equations.- 14.4 Multiscaling Methods.- 14.5 Boundary Layers.- 14.6 Remarks.- 14.7 References.- 15 Wave Propagation.- 15.0 Introduction.- 15.1 General Definitions and Classification of Waves.- 15.2 Physical Systems and Their Classification.- 15.3 Simple Waves: Nondispersive, Nondiffusive.- 15.4 Dispersive Waves.- 15.5 Diffusive Waves.- 15.6 References and Bibliography.- 16 Matrices and Linear Algebra.- 16.1 Preliminary Considerations.- 16.2 Determinants.- 16.3 Vector Spaces and Linear Transformation.- 16.4 Matrices.- 16.5 Linear System of Equations.- 16.6 Eigenvalues and the Jordan Normal Form.- 16.7 Estimates and Determination of Eigenvalues.- 16.8 Norms.- 16.9 Hermitian Forms and Matrices.- 16.10 Matrices with Real Elements.- 16.11 Generalized Inverse.- 16.12 Commuting Matrices.- 16.13 Compound Matrices.- 16.14 Handling Large Sparse Matrices.- 16.15 References.- 17 Functional Approximation.- 17.0 Introduction.- 17.1 Norms and Related Measures of Error.- 17.2 Relationship between Approximation on a Continuum and on a Discrete Point Set.- 17.3 Existence of Best Approximations.- 17.4 L2 or Least-Mean-Square Approximation.- 17.5 Theory of Chebyshev Approximation.- 17.6 Chebyshev Approximation Methods Based on Characterization Properties.- 17.7 Use of Linear Programming in Chebyshev Approximation.- 17.8 L1 Approximation.- 17.9 Piecewise Approximation without Continuity at the Joints.- 17.10 Approximation by Splines and Related Smooth Piecewise Functions.- 17.11 References.- 18 Numerical Analysis.- 18.0 Introduction.- 18.1 General Information on Error Analysis.- 18.2 Linear Equation Systems.- 18.3 Eigenvalue and Lambda-Matrix Problems.- 18.4 Approximation and Interpolation.- 18.5 Quadrature and Integral Equations.- 18.6 Ordinary Differential Equations.- 18.7 Nonlinear Functions of One Variable.- 18.8 Nonlinear Equation Systems and Optimization.- 18.9 Miscellaneous Topics.- 18.10 References and Bibliography.- 19 Mathematical Models and Their Formulation.- 19.1 Mathematical Modeling.- 19.2 Groping in the Dark.- 19.3 From the Simple to the Elaborate.- 19.4 Try a Different Formulation.- 19.5 Linearize with Care.- 19.6 Stepping Beyond Reality.- 19.7 Why Reinvent the Wheel?.- 19.8 Better Robust Than Realistic.- 19.9 References.- 20 Optimization Techniques.- 20.1 Introduction.- 20.2 Parameter Optimization.- 20.3 Dynamic Optimization, Neccessary Conditions.- 20.4 Extremal Fields, Sufficiency Conditions.- 20.5 Computational Techniques.- 20.6 Elements of Game Theory.- 20.7 References.- 21 Probability and Statistics.- 21.0 Introduction.- 21.1 Probability Spaces.- 21.2 Random Vectors and Random Variables.- 21.3 Descriptive Statistics.- 21.4 Statistical Inference.- 21.5 The General Linear Model.- 21.6 Some Other Techniques of Multivariate Analysis.- 21.7 Parametric, Nonparametric, and Distribution-Free Statistical Tests.- 21.8 Bayesian Statistics and Decision Theory.- 21.9 Concluding Remarks.- 21.10 References.