Handbook of Applied Mathematics: Selected Results and Methods

Most of the topics in applied mathematics dealt with in this handbook can be grouped rather loosely under the term analysis. They involve results and techniques which experience has shown to be of utility in a very broad variety of applications. Although care has been taken to collect certain basic results in convenient form, it is not the purpose of this handbook to duplicate the excellent collections of tables and formulas available in the National Bureau of Standards Handbook of Mathematical Functions (AMS Series 55, U.S. Government Printing Office) and in the references given therein. Rather, the emphasis in the present handbook is on technique, and we are indeed fortunate that a number of eminent applied mathe­ maticians have been willing to share with us their interpretations and experiences. To avoid the necessity of frequent and disruptive cross-referencing, it is expected that the reader will make full use of the index. Moreover, each chapter has been made as self-sufficient as is feasible. This procedure has resulted in occasional duplication, but as compensation for this the reader may appreciate the availability of different points of view concerning certain topics of current interest. As editor, I would like to express my appreciation to the contributing authors, to the reviewers, to the editorial staff of the publisher, and to the many secretaries and typists who have worked on the manuscript; without the partnership of all of these people, this handbook would not have been possible.

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.