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Index
Cover
Title Page
Copyright Page
The Authors
Contents
Foreword to the 1961 Edition
Preface to the 1961 Edition
Introduction to the 1961 Edition
Part 1. Mathematical Methods
1 From Delta Functions to Distributions BY ARTHUR ERDÉLYI
1.1 Introduction
Delta Functions and Other Generalized Functions
1.2 The Delta Function
1.3 Other Generalized Functions
Mikusiński’s Theory of Operational Calculus and Generalized Functions
1.4 The Definition of Operators
1.5 Differential and Integral Operators
1.6 Limits of Convolution Quotients
1.7 Operator Functions
1.8 Exponential Functions
1.9 The Diffusion Equation
1.10 Extensions and Other Theories
Distributions
1.11 Testing Functions
1.12 The Definition of Distributions
1.13 Operations with Distributions
1.14 Convergence of Distributions
1.15 Further Properties of Distributions
Applications and Extensions
1.16 Application to Fourier Transforms
1.17 Application to Differential Equations
1.18 Extensions and Alternative Theories
2 Operational Methods for Separable Differential Equations BY BERNARD FRIEDMAN
2.1 Introduction
2.2 Heaviside Theory
2.3 Domain of an Operator
2.4 Linear Operators
2.5 Functions of Operators
2.6 Eigenfunctions and Self-adjoint Operators
2.7 Spectral Representation
2.8 A Partial Differential Equation
2.9 Types of Spectral Representation
2.10 Conclusion
3 Integral Transforms BY JOHN W. MILES
3.1 Introduction
Inversion Formulas and Transform Pairs
3.2 Fourier’s Integral Formulas
3.3 Fourier Transform
3.4 Fourier Cosine and Sine Transforms
3.5 Laplace Transform
3.6 Mellin Transform
3.7 Multiple Fourier Transforms
3.8 Hankel Transforms
The Laplace Transform
3.9 Introduction
3.10 Transforms of Derivatives
3.11 Heaviside’s Shifting Theorem
3.12 Convolution Theorem
3.13 Inversion Procedures
3.14 A Problem in Wave Motion
3.15 A Problem in Heat Conduction
3.16 A Problem in Supersonic Flow
Fourier Transforms
3.17 Introduction
3.18 Transforms of Derivatives
3.19 Application to a Semi-infinite Domain
3.20 Initial-value Problem for One-dimensional Wave Equation
Hankel Transforms
3.21 Introduction
3.22 Problem of an Oscillating Piston
Finite Fourier Transforms
3.23 Introduction
3.24 Finite Cosine and Sine Transforms
3.25 A Problem in Wave Motion
3.26 Conclusion
4 Semigroup Methods in the Theory of Partial Differential Equations BY RALPH S. PHILLIPS
4.1 Introduction
4.2 Semigroups of Operators on Finite-dimensional Spaces
4.3 Hilbert Space
4.4 Semigroups of Operators on a Hilbert Space
4.5 Hyperbolic Systems of Partial Differential Equations
4.6 Maximal Dissipative Operators
4.7 Parabolic Partial Differential Equations
5 Asymptotic Formulas and Series BY J. BARKLEY ROSSER
5.1 Introduction
5.2 Definitions
5.3 Integration by Parts
5.4 The Generalized Watson’s Lemma
5.5 Asymptotic Solution of Differential Equations
5.6 Other Methods of Deriving Asymptotic Series
5.7 Eulerizing
5.8 Continued Fractions
5.9 Laplace’s Method
5.10 The Method of Stationary Phase
5.11 The Method of Steepest Descent
5.12 Further Use of Integration by Parts
Part 2. Statistical and Scheduling Studies
6 Chance Processes and Fluctuations BY WILLIAM FELLER
6.1 Introduction
Sums of Random Variables
6.2 Cumulative Effects
6.3 The Simplest Random-walk Model
6.4 The Fokker-Planck Equation
6.5 Example
6.6 Generalizations
6.7 The “Ruin” Problem
Queueing Problems
6.8 Holding and Waiting Times; Discipline
6.9 Random-walk Model; the Differential Equations
6.10 Steady State
6.11 Busy Periods
6.12 Fluctuations in the Individual Process vs. Ensemble Averages
6.13 The Example of D. G. Kendall’s Taxicab Stand
7 Information Theory BY DAVID BLACKWELL
7.1 Introduction
7.2 An Example
7.3 Entropy
7.4 Capacity of a Channel
7.5 The Fundamental Theorem
7.6 Multistate Channels
7.7 Entropy of a Process; Capacity of Finite-state Channels
8 The Mathematical Theory of Control Processes BY RICHARD BELLMAN
8.1 Introduction
Determinate Control Processes
8.2 The Calculus of Variations
8.3 A Catalogue of Catastrophes
8.4 Quadratic Criteria and Linear Equations
8.5 Linear Criteria and Linear Constraints
8.6 Nonlinear Criteria and Constraints
8.7 Implicit Functionals
8.8 Dynamic Programming
8.9 Trajectories
8.10 Computational Aspects
Stochastic Control Processes and Game Theory
8.11 Stochastic Effects
8.12 Games against Nature
8.13 Pursuit Processes
8.14 Analytic Techniques
Adaptive Control Processes
8.15 Adaptive Systems
8.16 Functional-equation Approach
8.17 Computational Aspects
An Illustrative Example
8.18 Formulation
8.19 Deterministic Case
8.20 Stochastic Case
8.21 Adaptive Case
9 Formulating and Solving Linear Programs BY GEORGE B. DANTZIG
9.1 Introduction
9.2 Formulating a Linear-programming Model
9.3 Building the Model
9.4 The Linear-programming Model Illustrated
9.5 Algebraic Statement of the Linear-programming Problem
9.6 Outline of the Simplex Method
9.7 Test for Optimal Feasible Solution
9.8 Improving a Nonoptimal Basic Feasible Solution
9.9 General Iterative Procedure
9.10 Finding an Initial Basic Feasible Solution
10 The Mathematical Theory of Inventory Processes BY SAMUEL KARLIN
10.1 Introduction
10.2 Factors of the Inventory Process
10.3 Cost Factors
10.4 The Nature of Demand
10.5 The Nature of Supply
10.6 The Structure of the Inventory Process
10.7 Classification of Inventory Models
10.8 Historical Inventory Models
10.9 The Literature of Inventory Theory
10.10 Deterministic Inventory Models
10.11 One-stage Stochastic Inventory Models
10.12 Optimal Policy for Dynamic Stochastic Inventory Problems
10.13 Model of Hydroelectric Generation with Stochastic Inflow
10.14 Steady-state Solution of Inventory Problems
10.15 Stationary Inventory Model
10.16 Inventory Model with a Random Supply
10.17 Stationary Distribution for a Model of Lagged Delivery
Part 3. Physical Phenomena
11 Monte Carlo Calculations in Problems of Mathematical Physics BY STANISLAW M. ULAM
11.1 Introduction
11.2 A Combinatorial Problem
11.3 Branching Processes
11.4 Multidimensional Branching Processes
11.5 Statistical Sampling Methods
11.6 Reactions in a Heavy Nucleus
11.7 The Petit Canonical Ensemble
11.8 Iterates of Transformations, Ergodic Properties, and Time Averages
12 Difference Equations and Functional Equations in Transmission-line Theory BY RAYMOND REDHEFFER
12.1 Introduction
The Algebraic Foundations
12.2 An Instructive Special Case
12.3 The Composition of Networks in General
12.4 Matrix Multiplication
12.5 Lossless Networks and the Reciprocity Theorem
12.6 Passive Networks
12.7 The Associated Linear Fractional Transformation
12.8 Another Characterization of Passive Networks
12.9 Fixed Points and Commutativity
12.10 Series of Obstacles; the Cascade Problem
12.11 Identical Networks in Cascade
Functional Equations
12.12 Homogeneous Anisotropic Media
12.13 Solution of the Equations
12.14 Application to the Cascade Problem
12.15 Interpretation of the Constants
12.16 Nonuniform Dielectric Media
12.17 Linearization
12.18 Conditions for a Passive Solution
12.19 Probability: a Reinterpretation
12.20 The Scattering Matrix
12.21 The Underlying Closure Principle
Transmission and Reflection Operators
12.22 Transmission, Reflection, and Scattering Matrices
12.23 The Star Product and Closure
12.24 The Norm and Energy Transfer
12.25 The Matching Problem
12.26 Further Discussion of Passive Networks
12.27 Inequalities for the Differential System
12.28 The Probability Scattering Matrix
12.29 A More General Interpretation
12.30 A Special Case and Examples
13 Characteristic-value Problems in Hydrodynamic and Hydromagnetic Theory BY SUBRAHMANYAN CHANDRASEKHAR
13.1 Introduction
13.2 The Rayleigh Criterion for the Stability of Inviscid Rotational Flow
13.3 Analytical Discussion of the Rayleigh Criterion
13.4 The Stability of Viscous Rotational Flow
13.5 On Methods of Solving Characteristic-value Problems in High-order Differential Equations
14 Applications of the Theory of Partial Differential Equations to Problems of Fluid Mechanics BY PAUL R. GARABEDIAN
14.1 Introduction
14.2 Cauchy’s Problem for a Hyperbolic Partial Differential Equation in Two Independent Variables
14.3 The Method of Finite Differences
14.4 Cauchy’s Problem in the Elliptic Case
14.5 Flow around a Bubble Rising under the Influence of Gravity
14.6 The Detached-shock Problem
15 The Numerical Solution of Elliptic and Parabolic Partial Differential Equations BY DAVID YOUNG
15.1 Introduction
15.2 Boundary-value Problems and the Method of Finite Differences
15.3 Point Iterative Methods
15.4 Peaceman-Rachford Iterative Method
15.5 Other Iterative Methods for Solving Elliptic Equations
15.6 Parabolic Equations—Forward-difference Method
15.7 The Crank-Nicolson Method
15.8 The Alternating-direction Method for Parabolic Equations Involving Two Space Variables
15.9 Illustrative Examples
15.10 The SPADE Project for the Development of a Computer Program for Solving Elliptic and Parabolic Equations
16 Circle, Sphere, Symmetrization, and Some Classical Physical Problems BY GEORGE PÓLYA
16.1 Introduction
The Heuristic Aspect
16.2 Observations
16.3 Conjectures
16.4 A Line of Inquiry
16.5 Plane
16.6 Space
16.7 Applications
The Key Idea of the Proof
16.8 Definition
16.9 From Surface Area to Dirichlet Integral
16.10 A Minor Remark
16.11 Symmetrization and Principal Frequency
16.12 Scope of the Proof
Additional Remarks
16.13 Alternative Symmetrization
16.14 Uniqueness
16.15 Where the Alternative Symmetrization Leaves No Alternative
16.16 One More Inequality Suggested by Observation
Name Index
Subject Index
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