Mathematical Handbook for Scientists and Engineers Definitions, Theorems, and Formulas for Reference and Review,9780486411477

Mathematical Handbook for Scientists and Engineers Definitions, Theorems, and Formulas for Reference and Review

by ;
Edition: 2nd
ISBN13: 9780486411477
ISBN10: 0486411478
Format: Paperback
Pub. Date: 2000-06-24
Publisher(s): Dover Publications
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Summary

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.

Table of Contents

Preface vii
Real and Complex Numbers, Elementary Algebra
1(28)
Introduction. The Real-number System
2(2)
Powers, Roots, Logarithms, and Factorials. Sum and Product Notation
4(3)
Complex Numbers
7(3)
Miscellaneous Formulas
10(2)
Determinants
12(3)
Algebraic Equations: General Theorems
15(4)
Factoring of Polynomials and Quotients of Polynomials. Partial Fractions
19(3)
Linear, Quadratic, Cubic, and Quartic Equations
22(2)
Systems of Simultaneous Equations
24(3)
Related Topics, References, and Bibliography
27(2)
Plane Analytic Geometry
29(28)
Introduction and Basic Concepts
30(7)
The Straight Line
37(2)
Relations Involving Points and Straight Lines
39(2)
Second-order Curves (Conic Sections)
41(7)
Properties of Circles, Ellipses, Hyperbolas, and Parabolas
48(5)
Higher Plane Curves
53(3)
Related Topics, References, and Bibliography
56(1)
Solid Analytic Geometry
57(26)
Introduction and Basic Concepts
58(9)
The Plane
67(2)
The Straight Line
69(1)
Relations Involving Points, Planes, and Straight Lines
70(4)
Quadric Surfaces
74(8)
Related Topics, References, and Bibliography
82(1)
Functions and Limits. Differential and Integral Calculus
83(62)
Introduction
85(1)
Functions
85(2)
Point Sets, Intervals, and Regions
87(3)
Limits, Continuous Functions, and Related Topics
90(5)
Differential Calculus
95(7)
Integrals and Integration
102(16)
Mean-value Theorems. Values of Indeterminate Forms. Weierstrass's Approximation Theorems
118(3)
Infinite Series, Infinite Products, and Continued Fractions
121(6)
Tests for the Convergence and Uniform Convergence of Infinite Series and Improper Integrals
127(4)
Representation of Functions by Infinite Series and Integrals. Power Series and Taylor's Expansion
131(3)
Fourier Series and Fourier Integrals
134(10)
Related Topics, References, and Bibliography
144(1)
Vector Analysis
145(23)
Introduction
146(1)
Vector Algebra
147(4)
Vector Calculus: Functions of a Scalar Parameter
151(2)
Scalar and Vector Fields
153(4)
Differential Operators
157(5)
Integral Theorems
162(2)
Specification of a Vector Field in Terms of Its Curl and Divergence
164(2)
Related Topics, References, and Bibliography
166(2)
Curvillinear Coordinate Systems
168(19)
Introduction
168(1)
Curvilinear Coordinate Systems
169(2)
Representation of Vectors in Terms of Components
171(3)
Orthogonal Coordinate Systems. Vector Relations in Terms of Orthogonal Components
174(3)
Formulas Relating to Special Orthogonal Coordinate Systems
177(1)
Related Topics, References, and Bibliography
177(10)
Functions of a Complex Variable
187(34)
Introduction
188(1)
Functions of a Complex Variable. Regions of the Complex-number Plane
188(4)
Analytic (Regular, Holomorphic) Functions
192(1)
Treatment of Multiple-valued Functions
193(3)
Integral Theorems and Series Expansions
196(2)
Zeros and Isolated Singularities
198(4)
Residues and Contour Integration
202(3)
Analytic Continuation
205(1)
Conformal Mapping
206(13)
Functions Mapping Specified Regions onto the Unit Circle
219(1)
Related Topics, References, and Bibliography
219(2)
The Laplace Transformation and Other Functional Transformations
221(22)
Introduction
222(1)
The Laplace Transformation
222(3)
Correspondence between Operations on Object and Result Functions
225(3)
Tables of Laplace-transform Pairs and Computation of Inverse Laplace Transforms
228(5)
``Formal'' Laplace Transformation of Impulse-function Terms
233(1)
Some Other Integral Transformations
233(4)
Finite Integral Transforms, Generating Functions, and z Transforms
237(3)
Related Topics, References, and Bibliography
240(3)
Ordinary Differential Equations
243(44)
Introduction
244(3)
First-order Equations
247(5)
Linear Differential Equations
252(13)
Linear Differential Equations with Constant Coefficients
265(12)
Nonlinear Second-order Equations
277(7)
Pfaffian Differential Equations
284(2)
Related Topics, References, and Bibliography
286(1)
Partial Differential Equations
287(43)
Introduction and Survey
288(2)
Partial Differential Equations of the First Order
290(12)
Hyperbolic, Parabolic, and Elliptic Partial Differential Equations. Characteristics
302(9)
Linear Partial Differential Equations of Physics. Particular Solutions
311(13)
Integral-transform Methods
324(4)
Related Topics, References, and Bibliography
328(2)
Maxima and Minima and Optimization Problems
330(43)
Introduction
321(11)
Maxima and Minima of Functions of One Real Variable
332(1)
Maxima and Minima of Functions of Two or More Real Variables
333(2)
Linear Programming, Games, and Related Topics
335(9)
Calculus of Variations. Maxima and Minima of Definite Integrals
344(2)
Extremals as Solutions of Differential Equations: Classical Theory
346(11)
Solution of Variation Problems by Direct Methods
357(1)
Control Problems and the Maximum Principle
358(11)
Stepwise-control Problems and Dynamic Programming
369(2)
Related Topics, References, and Bibliography
371(2)
Definition of Mathematical Models: Modern (Abstract) Algebra and Abstract Spaces
373(29)
Introduction
374(4)
Algebra of Models with a Single Defining Operation: Groups
378(4)
Algebra of Models with Two Defining Operations: Rings, Fields, and Integral Domains
382(2)
Models Involving More Than One Class of Mathematical Objects: Linear Vector Spaces and Linear Algebras
384(2)
Models Permitting the Definition of Limiting Processes: Topological Spaces
386(5)
Order
391(1)
Combination of Models: Direct Products, Product Spaces, and Direct Sums
392(1)
Boolean Algebras
393(7)
Related Topics, References, and Bibliography
400(2)
Matrices. Quadratic and Hermitian Forms
402(29)
Introduction
403(1)
Matrix Algebra and Matrix Calculus
403(7)
Matrices with Special Symmetry Properties
410(2)
Equivalent Matrices. Eigenvalues, Diagonalization, and Related Topics
412(4)
Quadratic and Hermitian Forms
416(4)
Matrix Notation for Systems of Differential Equations (State Equations). Perturbations and Lyapunov Stability Theory
420(10)
Related Topics, References, and Bibliography
430(1)
Linear Vector Spaces and Linear Transformations (Linear Operators). Representation of Mathematical Models in Terms of Matrices
431(53)
Introduction. Reference Systems and Coordinate Transformations
433(2)
Linear Vector Spaces
435(4)
Linear Transformations (Linear Operations)
439(2)
Linear Transformations of a Normed or Unitary Vector Space into Itself. Hermitian and Unitary Transformations (Operators)
441(6)
Matrix Representation of Vectors and Linear Transformations (Operators)
447(2)
Change of Reference System
449(3)
Representation of Inner Products. Orthonormal Bases
452(5)
Eigenvector and Eigenvalues of Linear Operators
457(10)
Group Representations and Related Topics
467(4)
Mathematical Description of Rotations
471(11)
Related Topics, References, and Bibliography
482(2)
Linear Integral Equations, Boundary-value Problems, and Eigenvalue Problems
484(49)
Introduction. Functional Analysis
486(1)
Functions as Vectors. Expansions in Terms of Orthogonal Functions
487(5)
Linear Integral Transformations and Linear Integral Equations
492(10)
Linear Boundary-value Problems and Eigenvalue Problems Involving Differential Equations
502(13)
Green's Functions. Relation of Boundary-value Problems and Eigenvalue Problems to Integral Equations
515(5)
Potential Theory
520(11)
Related Topics, References, and Bibliography
531(2)
Representation of Mathematical Models: Tensor Algebra and Analysis
533(28)
Introduction
534(3)
Absolute and Relative Tensors
537(3)
Tensor Algebra: Definition of Basic Operations
540(3)
Tensor Algebra: Invariance of Tensor Equations
543(1)
Symmetric and Skew-symmetric Tensors
543(2)
Local Systems of Base Vectors
545(1)
Tensors Defined on Riemann Spaces. Associated Tensors
546(3)
Scalar Products and Related Topics
549(2)
Tensors of Rank Two (Dyadics) Defined on Riemann Spaces
551(1)
The Absolute Differential Calculus. Covariant Differentiation
552(8)
Related Topics, References, and Bibliography
560(1)
Differential Geometry
561(24)
Curves in the Euclidean Plane
562(3)
Curves in Three-dimensional Euclidean Space
565(4)
Surfaces in Three-dimensional Euclidean Space
569(10)
Curved Spaces
579(5)
Related Topics, References, and Bibliography
584(1)
Probability Theory and Random Processes
585(79)
Introduction
587(1)
Definition and Representation of Probability Models
588(5)
One-dimensional Probability Distributions
593(9)
Multidimensional Probability Distributions
602(12)
Functions of Random Variables. Change of Variables
614(6)
Convergence in Probability and Limit Theorems
620(3)
Special Techniques for Solving Probability Problems
623(2)
Special Probability Distributions
625(12)
Mathematical Description of Random Processes
637(4)
Stationary Random Processes. Correlation Functions and Spectral Densities
641(9)
Special Classes of Random Processes. Examples
650(9)
Operations on Random Processes
659(3)
Related Topics, References, and Bibliography
662(2)
Mathematical Statistics
664(51)
Introduction to Statistical Methods
665(3)
Statistical Description. Definition and Computation of Randomsample Statistics
668(6)
General-purpose Probability Distributions
674(2)
Classical Parameter Estimation
676(4)
Sampling Distributions
680(6)
Classical Statistical Tests
686(11)
Some Statistics, Sampling Distributions, and Tests for Multivariate Distributions
697(6)
Random-process Statistics and Measurements
703(5)
Testing and Estimation with Random Parameters
708(5)
Related Topics, References, and Bibliography
713(2)
Numerical Calculations and Finite Differences
715(89)
Introduction
718(1)
Numerical Solution of Equations
719(10)
Linear Simultaneous Equations, Matrix Inversion, and Matrix Eigenvalue Problems
729(8)
Finite Differences and Difference Equations
737(9)
Approximation of Functions by Interpolation
746(9)
Approximation by Orthogonal Polynomials, Truncated Fourier Series, and Other Methods
755(15)
Numerical Differentiation and Integration
770(7)
Numerical Solution of Ordinary Differential Equations
777(8)
Numerical Solution of Boundary-value Problems, Partial Differential Equations, and Integral Equations
785(12)
Monte-Carlo Techniques
797(3)
Related Topics, References, and Bibliography
800(4)
Special Functions
804(77)
Introduction
806(1)
The Elementary Transcendental Functions
806(12)
Some Functions Defined by Transcendental Integrals
818(4)
The Gamma Function and Related Functions
822(2)
Binomial Coefficients and Factorial Polynomials. Bernoulli Polynomials and Bernoulli Numbers
824(3)
Elliptic Functions, Elliptic Integrals, and Related Functions
827(21)
Orthogonal Polynomials
848(9)
Cylinder Functions, Associated Legendre Functions, and Spherical Harmonics
857(17)
Step Functions and Symbolic Impulse Functions
874(6)
References and Bibliography
880(1)
Appendix A. Formulas Describing Plane Figures and Solids 881(4)
Appendix B. Plane and Spherical Trigonometry 885(9)
Appendix C. Permutations, Combinations, and Related Topics 894(6)
Appendix D. Tables of Fourier Expansions and Laplace-transform Pairs 900(25)
Appendix E. Integrals, Sums, Infinite Series and Products, and Continued Fractions 925(64)
Appendix F. Numerical Tables 989(101)
Squares
990(3)
Logarithms
993(17)
Trigonometric Functions
1010(8)
Exponential and Hyperbolic Functions
1018(7)
Natural Logarithms
1025(2)
Sine Integral
1027(1)
Cosine Integral
1028(1)
Exponential and Related Integrals
1029(4)
Complete Elliptic Integrals
1033(1)
Factorials and Their Reciprocals
1034(1)
Binomial Coefficients
1034(1)
Gamma and Factorial Functions
1035(2)
Bessel Functions
1037(23)
Legendre Polynomials
1060(1)
Error Function
1061(1)
Normal-distribution Areas
1062(1)
Normal-curve Ordinates
1063(1)
Distribution of t
1064(1)
Distribution of x2
1065(1)
Distribution of F
1066(4)
Random Numbers
1070(5)
Normal Random Numbers
1075(5)
sin x/x
1080(9)
Chebyshev Polynomials
1089(1)
Glossary of Symbols and Notations 1090(7)
Index 1097

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