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Mathematical Handbook for Scientists & E 2ND Edition


Mathematical Handbook for Scientists & E 2ND Edition Cover


Synopses & Reviews

Publisher Comments:

A reliable source of definitions, theorems, and formulas, this authoritative handbook provides convenient access to information from every area of mathematics. Coverage includes Fourier transforms, Z transforms, linear and nonlinear programming, calculus of variations, random-process theory, special functions, combinatorial analysis, numerical methods, game theory, and much more.


Convenient access to information from every area of mathematics: Fourier transforms, Z transforms, linear and nonlinear programming, calculus of variations, random-process theory, special functions, combinatorial analysis, game theory, much more.

Table of Contents


Chapter 1. Real and Complex Numbers. Elementary Algebra.

1.1. Introduction. The Real-number System

1.2. "Powers, Roots, Logarithms, and Factorials. Sum and Product Notation"

1.3. Complex Numbers

1.4. Miscellaneous Formulas

1.5. Determinants

1.6. Algebraic Equations: General Theorems

1.7. Factoring of Polynomials and Quotients of Polynomials. Partial Fractions

1.8. "Linear, Quadratic, Cubic, and Quartic Equations"

1.9. Systems of Simultaneous Equations

1.10. "Related Topics, References, and Bibliography"

Chapter 2. Plane Analytic Geometry

2.1. Introduction and Basic Concepts

2.2. The Straight Line

2.3. Relations Involving Points and Straight Lines

2.4. Second-order Curves (Conic Sections)

2.5. "Properties of Circles, Ellipses, Hyperbolas, and Parabolas"

2.6. Higher Plane Curves

2.7. "Related Topics, References, and Bibliography"

Chapter 3. Solid Analytic Geometry

3.1. Introduction and Basic Concepts

3.2. The Plane

3.3. The Straight Line

3.4. "Relations Involving Points, Planes, and Straight Lines"

3.5. Quadric Surfaces

3.6. "Related Topics, References, and Bibliography"

Chapter 4. Functions and Limits. Differential and Integral Calculus

4.1. Introduction

4.2. Functions

4.3. "Point Sets, Intervals, and Regions"

4.4. "Limits, Continuous Functions, and Related Topics"

4.5. Differential Calculus

4.6. Integrals and Integration

4.7. Mean-value Theorems. Values of Indeterminate Forms. Weierstrass's Approximation Theorems.

4.8. "Infinite Series, Infinite Products, and Continued Fractions"

4.9. Tests for the Convergence and Uniform Convergence of Infinite Series and Improper Integrals

4.10. Representation of Functions by Infinite Series and Integrals. Power Series and Taylor's Expansion

4.11. Fourier Series and Fourier Integrals

4.12. "Related Topics, References, and Bibliography"

Chapter 5. Vector Analysis

5.1. Introduction

5.2. Vector Algebra

5.3. Vector Calculus: Functions of Scalar Parameter

5.4. Scalar and Vector Fields

5.5. Differential Operators

5.6. Integral Theorems

5.7. Specification of a Vector Field in Terms of Its Curl and Divergence

5.8. "Related Topics, References, and Bibliography"

Chapter 6. Curvilinear Coordinate Systems

6.1. Introduction

6.2. Curvilinear Coordinate Systems

6.3. Representation of Vectors in Terms of Components

6.4. Orthogonal Coordinate Systems. Vector Relations in Terms of Orthogonal Components

6.5. Formulas Relating to Special Orthogonal Coordinate Systems

6.6. "Related Topics, References, and Bibliography"

Chapter 7. Functions of a Complex Variable

7.1. Introduction

7.2. Functions of a Complex Variable. Regions of the Complex-number Plane

7.3. "Analytic (Regular, Holomorphic) Functions"

7.4. Treatment of Multiple-valued Functions

7.5. Integral Theorems and Series Expansions

7.6. Zeros and Isolated Singularities

7.7. Residues and Contour Integration

7.8. Analytic Continuation

7.9. Conformal Mapping

7.10. Functions Mapping Specified Regions onto the Unit Circle

7.11. "Related Topics, References, and Bibliography"

Chapter 8. The Laplace Transformation and Other Functional Transformations

8.1. Introduction

8.2. The Laplace Transformation

8.3. Correspondence between Operations on Object and Result Functions

8.4. Table of Laplace-transform Pairs and Computation of Inverse Laplace Transforms

8.5. "Formal" Laplace Transformation of Impulse-function Terms"

8.6. Some Other Integral Transformations

8.7. "Finite Integral Transforms, Generating Functions, and z Transforms"

8.8. "Related Topics, References, and Bibliography"

Chapter 9. Ordinary Differential Equations

9.1. Introduction

9.2. First-order Equations

9.3. Linear Differential Equations

9.4. Linear Differential Equations with Constant Coefficients

9.5. Nonlinear Second-order Equations

9.6. Pfaffian Differential Equations

9.7. "Related Topics, References, and Bibliography"

Chapter 10. Partial Differential Equations

10.1. Introduction and Survey

10.2. Partial Differential Equations of the First Order

10.3. "Hyperbolic, Parabolic, and Elliptic Partial Differential Equations. Characteristics."

10.4. Linear Partial Differential Equations of Physics. Particular Solutions.

10.5. Integral-transform Methods

10.6. "Related Topics, References, and Bibliography"

Chapter 11. Maxima and Minima and Optimization Problems

11.1. Introduction

11.2. Maxima and Minima of Functions of One Real Variable

11.3. Maxima and Minima of Functions of Two or More Real Variables

11.4. "Linear Programming, Games, and Related Topics"

11.5. Calculus of Variations. Maxima and Minima of Definite Integrals

11.6. Extremals as Solutions of Differential Equations: Classical Theory

11.7. Solution of Variation Problems by Direct Methods

11.8. Control Problems and the Maximum Principle

11.9. Stepwise-control Problems and Dynamic Programming

11.10. "Related Topics, References, and Bibliography"

Chapter 12. Definition of Mathematical Models: Modern (Abstract) Algebra and Abstract Spaces

12.1. Introduction

12.2. Algebra of Models with a Single Defining Operation: Groups

12.3. "Algebra of Models with Two Defining Operations: Rings, Fields, and Integral Domains"

12.4. Models Involving More Than One Class of Mathematical Objects: Linear Vector Spaces and Linear Algebras

12.5. Models Permitting the Definition of Limiting Processes: Topological Spaces

12.6. Order

12.7. "Combination of Models: Direct Products, Product Spaces, and Direct Sums"

12.8. Boolean Algebras

12.9. "Related Topics, References, and Bibliography"

Chapter 13. Matrices. Quadratic and Hermitian Forms

13.1. Introduction

13.2. Matrix Algebra and Matrix Calculus

13.3. Matrices with Special Symmetry Properties

13.4. "Equivalent Matrices. Eigenvalues, Diagonalization, and Related Topics"

13.5. Quadratic and Hermitian Forms

13.6. Matrix Notation for Systems of Differential Equations (State Equations). Perturbations and Lyapunov Stability Theory

13.7. "Related Topics, References, and Bibliography"

Chapter 14. Linear Vector Spaces and Linear Transformations (Linear Operators). Representation of Mathematical Models in Terms of Matrices

14.1. Introduction. Reference Systems and Coordinate Transformations

14.2. Linear Vector Spaces

14.3. Linear Transformations (Linear Operators)

14.4. Linear Transformations of a Normed or Unitary Vector Space into Itself. Hermitian and Unitary Transformations (Operators)

14.5. Matrix Representation of Vectors and Linear Transformations (Operators)

14.6. Change of Reference System

14.7. Representation of Inner Products. Orthonormal Bases

14.8. Eigenvectors and Eigenvalues of Linear operators

14.9. Group Representations and Related Topics

14.10. Mathematical Description of Rotations

14.11. "Related Topics, References, and Bibliography"

"Chapter 15. Linear Integral Equations, Boundary-value Problems, and Eigenvalue Problems"

15.1. Introduction. Functional Analysis

15.2. Functions as Vectors. Expansions in Terms of Orthogonal Functions

15.3. Linear Integral Transformations and Linear Integral Equations

15.4. Linear Boundary-value Problems and Eigenvalue Problems Involving Differential Equations

15.5. Green's Functions. Relation of Boundary-value Problems and Eigenvalue Problems to Integral Equations

15.6. Potential Theory

15.7. "Related Topics, References, and Bibliography"

Chapter 16. Representation of Mathematical Models: Tensor Algebra and Analysis

16.1. Introduction

16.2. Absolute and Relative Tensors

16.3. Tensor Algebra: Definition of Basic Operators

16.4. Tensor Algebra: Invariance of Tensor Equations

16.5. Symmetric and Skew-Symmetric Tensors

16.6. Local Systems of Base Vectors

16.7. Tensors Defined on Riemann Spaces. Associated Tensors

16.8. Scalar Products and Related Topics

16.9. Tensors of Rank Two (Dyadics) Defined on Riemann Spaces

16.10. The Absolute Differential Calculus. Covariant Differentiation

16.11. "Related Topics, References, and Bibliography"

Chapter 17. Differential Geometry

17.1. Curves in the Euclidean Plane

17.2. Curves in the Three-dimensional Euclidean Space

17.3. Surfaces in Three-dimensional Euclidean Space

17.4. Curved Spaces

17.5. "Related Topics, References, and Bibliography"

Chapter 18. Probability Theory and Random Processes

18.1. Introduction

18.2. Definition and Representation of Probability Models

18.3. One-dimensional Probability Distributions

18.4. Multidimensional Probability Distributions

18.5. Functions of Random Variables. Change of Variables

18.6. Convergence in Probability and Limit Theorems

18.7. Special Techniques for Solving Probability Theorems

18.8. Special Probability Distributions

18.9. Mathematical Description of Random Processes

18.10. Stationary Random Processes. Correlation Functions and Spectral Densities

18.11. Special Classes of Random Processes. Examples

18.12. Operations on Random Processes

18.13. "Related Topics, References, and Bibliography"

Chapter 19. Mathematical Statistics

19.1. Introduction to Statistical Methods

19.2. Statistical Description. Definition and Computation of Random-sample Statistics

19.3. General-purpose Probability Distributions

19.4. Classical Parameter Estimation

19.5. Sampling Distributions

19.6. Classical Statistical Tests

19.7. "Some Statistics, Sampling Distributions, and Tests for Multivariate Distributions"

19.8. Random-process Statistics and Measurements

19.9. Testing and Estimation with Random Parameters

19.10. "Related Topics, References, and Bibliography"

Chapter 20. Numerical Calculations and Finite Differences

20.1. Introduction

20.2. Numerical Solution of Equations

20.3. "Linear Simultaneous Equations, Matrix Inversion, and Matrix Eigenvalue Problems"

20.4. Finite Differences and Difference Equations

20.5. Approximation of Functions by Interpolation

20.6. "Approximation by Orthogonal Polynomials, Truncated Fourier Series, and Other Methods"

20.7. Numerical Differentiation and Integration

20.8. Numerical Solution of Ordinary Differential Equations

20.9. "Numerical Solution of Boundary-value Problems, Partial Differential Equations, and Integral Equations"

20.10. Monte-Carlo Techniques

20.11. "Related Topics, References, and Bibliography"

Chapter 21. Special Functions

21.1. Introduction

21.2. The Elementary Transcendental Functions

21.3. Some Functions Defined by Transcendental Integrals

21.4. The Gamma Function and Related Functions

21.5. Binomial Coefficients and Factorial Polynomials. Bernoulli Polynomials and Bernoulli Numbers.

21.6. "Elliptic Functions, Elliptic Integrals, and Related Functions"

21.7. Orthogonal Polynomials

21.8. "Cylinder Functions, Associated Legendre Functions, and Spherical Harmonics"

21.9. Step Functions and Symbolic Impulse Functions

21.10. References and Bibliography

Appendix A. Formulas Describing Plane Figures and Solids

Appendix B. Plane and Spherical Trigonometry

"Appendix C. Permutations, Combinations, and Related Topics"

Appendix D. Tables of Fourier Expansions and Laplace-transform Pairs

"Appendix E. Integrals, Sums, Infinite Series and Products, and Continued Fractions"

Appendix F. Numerical Tables



  Trigonometric Functions

  Exponential and Hyperbolic Functions

  Natural Logarithms

  Sine Integral

  Cosine Integral

  Exponential and Related Integrals

  Complete Elliptic Integrals

  Factorials and Their Reciprocals

  Binomial Coefficients

  Gamma and Factorial Functions

  Bessel Functions

  Legendre Polynomials

  Error Function

  Normal-distribution Areas

  Normal-curve Ordinates

  Distribution of t

  Distribution of x²

  Distribution of F

  Random Numbers

  Normal Random Numbers

  sin x/x

  Chebyshev Polynomials

Glossary of Symbols and Notations


Product Details

Korn, Granino Arthur
Korn, Theresa M.
Korn, Granino Arthur
Korn, Granino A.
Dover Publications
Mineola, N.Y.
Engineering - Civil
Edition Number:
Dover ed.
Edition Description:
Trade Paper
Dover Civil and Mechanical Engineering
Series Volume:
Publication Date:
8.5 x 5.38 in 2.63 lb

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Convenient access to information from every area of mathematics: Fourier transforms, Z transforms, linear and nonlinear programming, calculus of variations, random-process theory, special functions, combinatorial analysis, game theory, much more.
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