Handbook of Mathematics for Engineers And Scientists

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ISBN: 9781584885023 | 1584885025
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Copyright: 11/27/2006

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The Handbook of Mathematics for Engineers and Scientists covers the main fields of mathematics and focuses on the methods used for obtaining solutions of various classes of mathematical equations that underlie the mathematical modeling of numerous phenomena and processes in science and technology. To accommodate different mathematical backgrounds, the preeminent authors outline the material in a simplified, schematic manner, avoiding special terminology wherever possible.Organized in ascending order of complexity, the material is divided into two parts. The first part is a coherent survey of the most important definitions, formulas, equations, methods, and theorems. It covers arithmetic, elementary and analytic geometry, algebra, differential and integral calculus, special functions, calculus of variations, and probability theory. Numerous specific examples clarify the methods for solving problems and equations. The second part provides many in-depth mathematical tables, including those of exact solutions of various types of equations.This concise, comprehensive compendium of mathematical definitions, formulas, and theorems provides the foundation for exploring scientific and technological phenomena.

Authors

xxv

Preface

xxvii

Main Notation

xxix

Part I. Definitions, Formulas, Methods, and Theorems

(1110)

Arithmetic and Elementary Algebra

(16)

Real Numbers

(2)

Integer Numbers

(1)

Real, Rational, and Irrational Numbers

(1)

Equalities and Inequalities. Arithmetic Operations. Absolute Value

(3)

Equalities and Inequalities

(1)

Addition and Multiplication of Numbers

(1)

Ratios and Proportions

(1)

Percentage

(1)

Absolute Value of a Number (Modulus of a Number)

(1)

Powers and Logarithms

(2)

Powers and Roots

(1)

Logarithms

(1)

Binomial Theorem and Related Formulas

(1)

Factorials. Binomial Coefficients. Binomial Theorem

(1)

Related Formulas

(1)

Arithmetic and Geometric Progressions. Finite Sums and Products

(2)

Arithmetic and Geometric Progressions

(1)

Finite Series and Products

(1)

Mean Values and Inequalities of General Form

(2)

Arithmetic Mean, Geometric Mean, and Other Mean Values. Inequalities for Mean Values

(1)

Inequalities of General Form

(1)

Some Mathematical Methods

(4)

Proof by Contradiction

(1)

Mathematical Induction

(1)

Proof by Counterexample

(1)

Method of Undetermined Coefficients

(1)

References for Chapter 1

(1)

Elementary Functions

(24)

Power, Exponential, and Logarithmic Functions

(5)

Power Function: y = xα

(2)

Exponential Function: y = ax

(1)

Logarithmic Function: y = loga x

(2)

Trigonometric Functions

(6)

Trigonometric Circle. Definition of Trigonometric Functions

(1)

Graphs of Trigonometric Functions

(2)

Properties of Trigonometric Functions

(3)

Inverse Trigonometric Functions

(4)

Definitions. Graphs of Inverse Trigonometric Functions

(3)

Properties of Inverse Trigonometric Functions

(1)

Hyperbolic Functions

(5)

Definitions. Graphs of Hyperbolic Functions

(2)

Properties of Hyperbolic Functions

(3)

Inverse Hyperbolic Functions

(4)

Definitions. Graphs of Inverse Hyperbolic Functions

(2)

Properties of Inverse Hyperbolic Functions

(1)

References for Chapter 2

(1)

Elementary Geometry

(34)

Plane Geometry

(16)

Triangles

(8)

Polygons

(5)

Circle

(3)

Solid Geometry

(11)

Straight Lines, Planes, and Angles in Space

(2)

Polyhedra

(4)

Solids Formed by Revolution of Lines

(5)

Spherical Trigonometry

(7)

Spherical Geometry

(1)

Spherical Triangles

(4)

References for Chapter 3

(2)

Analytic Geometry

(78)

Points, Segments, and Coordinates on Line and Plane

(7)

Coordinates on Line

(1)

Coordinates on Plane

(3)

Points and Segments on Plane

(3)

Curves on Plane

(5)

Curves and Their Equations

(4)

Main Problems of Analytic Geometry for Curves

(1)

Straight Lines and Points on Plane

(8)

Equations of Straight Lines on Plane

(4)

Mutual Arrangement of Points and Straight Lines

(4)

Second-Order Curves

(16)

Circle

(1)

Ellipse

(3)

Hyperbola

101

(3)

Parabola

104

(3)

Transformation of Second-Order Curves to Canonical Form

107

(6)

Coordinates, Vectors, Curves, and Surfaces in Space

113

(11)

Vectors. Cartesian Coordinate System

113

(1)

Coordinate Systems

114

(6)

Vectors. Products of Vectors

120

(3)

Curves and Surfaces in Space

123

(1)

Line and Plane in Space

124

(19)

Plane in Space

124

(7)

Line in Space

131

(4)

Mutual Arrangement of Points, Lines, and Planes

135

(8)

Quadric Surfaces (Quadrics)

143

(12)

Quadrics (Canonical Equations)

143

(5)

Quadrics (General Theory)

148

(5)

References for Chapter 4

153

(2)

Algebra

155

(80)

Polynomials and Algebraic Equations

155

(12)

Polynomials and Their Properties

155

(2)

Linear and Quadratic Equations

157

(1)

Cubic Equations

158

(1)

Fourth-Degree Equation

159

(2)

Algebraic Equations of Arbitrary Degree and Their Properties

161

(6)

Matrices and Determinants

167

(20)

Matrices

167

(8)

Determinants

175

(5)

Equivalent Matrices. Eigenvalues

180

(7)

Linear Spaces

187

(5)

Concept of a Linear Space. Its Basis and Dimension

187

(3)

Subspaces of Linear Spaces

190

(1)

Coordinate Transformations Corresponding to Basis Transformations in a Linear Space

191

(1)

Euclidean Spaces

192

(5)

Real Euclidean Space

192

(3)

Complex Euclidean Space (Unitary Space)

195

(1)

Banach Spaces and Hilbert Spaces

196

(1)

Systems of Linear Algebraic Equations

197

(7)

Consistency Condition for a Linear System

197

(1)

Finding Solutions of a System of Linear Equations

198

(6)

Linear Operators

204

(9)

Notion of a Linear Operator. Its Properties

204

(4)

Linear Operators in Matrix Form

208

(1)

Eigenvectors and Eigenvalues of Linear Operators

209

(4)

Bilinear and Quadratic Forms

213

(12)

Linear and Sesquilinear Forms

213

(1)

Bilinear Forms

214

(2)

Quadratic Forms

216

(3)

Bilinear and Quadratic Forms in Euclidean Space

219

(1)

Second-Order Hypersurfaces

220

(5)

Some Facts from Group Theory

225

(10)

Groups and Their Basic Properties

225

(3)

Transformation Groups

228

(2)

Group Representations

230

(3)

References for Chapter 5

233

(2)

Limits and Derivatives

235

(38)

Basic Concepts of Mathematical Analysis

235

(15)

Number Sets. Functions of Real Variable

235

(2)

Limit of a Sequence

237

(3)

Limit of a Function. Asymptotes

240

(2)

Infinitely Small and Infinitely Large Functions

242

(1)

Continuous Functions. Discontinuities of the First and the Second Kind

243

(2)

Convex and Concave Functions

245

(1)

Functions of Bounded Variation

246

(3)

Convergence of Functions

249

(1)

Differential Calculus for Functions of a Single Variable

250

(13)

Derivative and Differential, Their Geometrical and Physical Meaning

250

(2)

Table of Derivatives and Differentiation Rules

252

(2)

Theorems about Differentiable Functions. L'Hospital Rule

254

(1)

Higher-Order Derivatives and Differentials. Taylor's Formula

255

(2)

Extremal Points. Points of Inflection

257

(2)

Qualitative Analysis of Functions and Construction of Graphs

259

(1)

Approximate Solution of Equations (Root-Finding Algorithms for Continuous Functions)

260

(3)

Functions of Several Variables. Partial Derivatives

263

(10)

Point Sets. Functions. Limits and Continuity

263

(1)

Differentiation of Functions of Several Variables

264

(3)

Directional Derivative. Gradient. Geometrical Applications

267

(2)

Extremal Points of Functions of Several Variables

269

(3)

Differential Operators of the Field Theory

272

(1)

References for Chapter 6

272

(1)

Integrals

273

(64)

Indefinite Integral

273

(13)

Antiderivative. Indefinite Integral and Its Properties

273

(1)

Table of Basic Integrals. Properties of the Indefinite Integral. Integration Examples

274

(2)

Integration of Rational Functions

276

(3)

Integration of Irrational Functions

279

(2)

Integration of Exponential and Trigonometric Functions

281

(2)

Integration of Polynomials Multiplied by Elementary Functions

283

(3)

Definite Integral

286

(31)

Basic Definitions. Classes of Integrable Functions. Geometrical Meaning of the Definite Integral

286

(1)

Properties of Definite Integrals and Useful Formulas

287

(2)

General Reduction Formulas for the Evaluation of Integrals

289

(1)

General Asymptotic Formulas for the Calculation of Integrals

290

(5)

Mean Value Theorems. Properties of Integrals in Terms of Inequalities. Arithmetic Mean and Geometric Mean of Functions

295

(4)

Geometric and Physical Applications of the Definite Integral

299

(2)

Improper Integrals with Infinite Integration Limit

301

(3)

General Reduction Formulas for the Calculation of Improper Integrals

304

(3)

General Asymptotic Formulas for the Calculation of Improper Integrals

307

(1)

Improper Integrals of Unbounded Functions

308

(2)

Cauchy-Type Singular Integrals

310

(2)

Stieltjes Integral

312

(2)

Square Integrable Functions

314

(1)

Approximate (Numerical) Methods for Computation of Definite Integrals

315

(2)

Double and Triple Integrals

317

(12)

Definition and Properties of the Double Integral

317

(2)

Computation of the Double Integral

319

(4)

Geometric and Physical Applications of the Double Integral

323

(1)

Definition and Properties of the Triple Integral

324

(1)

Computation of the Triple Integral. Some Applications. Iterated Integrals and Asymptotic Formulas

325

(4)

Line and Surface Integrals

329

(8)

Line Integral of the First Kind

329

(1)

Line Integral of the Second Kind

330

(2)

Surface Integral of the First Kind

332

(1)

Surface Integral of the Second Kind

333

(1)

Integral Formulas of Vector Calculus

334

(1)

References for Chapter 7

335

(2)

Series

337

(30)

Numerical Series and Infinite Products

337

(11)

Convergent Numerical Series and Their Properties. Cauchy's Criterion

337

(1)

Convergence Criteria for Series with Positive (Nonnegative) Terms

338

(3)

Convergence Criteria for Arbitrary Numerical Series. Absolute and Conditional Convergence

341

(2)

Multiplication of Series. Some Inequalities

343

(1)

Summation Methods. Convergence Acceleration

344

(2)

Infinite Products

346

(2)

Functional Series

348

(2)

Pointwise and Uniform Convergence of Functional Series

348

(1)

Basic Criteria of Uniform Convergence. Properties of Uniformly Convergent Series

349

(1)

Power Series

350

(7)

Radius of Convergence of Power Series. Properties of Power Series

350

(2)

Taylor and Maclaurin Power Series

352

(2)

Operations with Power Series. Summation Formulas for Power Series

354

(3)

Fourier Series

357

(6)

Representation of 2π-Periodic Functions by Fourier Series. Main Results

357

(2)

Fourier Expansions of Periodic, Nonperiodic, Odd, and Even Functions

359

(2)

Criteria of Uniform and Mean-Square Convergence of Fourier Series

361

(1)

Summation Formulas for Trigonometric Series

362

(1)

Asymptotic Series

363

(4)

Asymptotic Series of Poincare Type. Formulas for the Coefficients

363

(1)

Operations with Asymptotic Series

364

(2)

References for Chapter 8

366

(1)

Differential Geometry

367

(32)

Theory of Curves

367

(19)

Plane Curves

367

(12)

Space Curves

379

(7)

Theory of Surfaces

386

(13)

Elementary Notions in Theory of Surfaces

386

(6)

Curvature of Curves on Surface

392

(3)

Intrinsic Geometry of Surface

395

(2)

References for Chapter 9

397

(2)

Functions of Complex Variable

399

(36)

Basic Notions

399

(20)

Complex Numbers. Functions of Complex Variable

399

(2)

Functions of Complex Variable

401

(18)

Main Applications

419

(16)

Conformal Mappings

419

(8)

Boundary Value Problems

427

(6)

References for Chapter 10

433

(2)

Integral Transforms

435

(18)

General Form of Integral Transforms. Some Formulas

435

(1)

Integral Transforms and Inversion Formulas

435

(1)

Residues. Jordan Lemma

435

(1)

Laplace Transform

436

(5)

Laplace Transform and the Inverse Laplace Transform

436

(1)

Main Properties of the Laplace Transform. Inversion Formulas for Some Functions

437

(3)

Limit Theorems. Representation of Inverse Transforms as Convergent Series and Asymptotic Expansions

440

(1)

Mellin Transform

441

(2)

Mellin Transform and the Inversion Formula

441

(1)

Main Properties of the Mellin Transform. Relation Among the Mellin, Laplace, and Fourier Transforms

442

(1)

Various Forms of the Fourier Transform

443

(3)

Fourier Transform and the Inverse Fourier Transform

443

(2)

Fourier Cosine and Sine Transforms

445

(1)

Other Integral Transforms

446

(7)

Integral Transforms Whose Kernels Contain Bessel Functions and Modified Bessel Functions

446

(2)

Summary Table of Integral Transforms. Areas of Application of Integral Transforms

448

(3)

References for Chapter 11

451

(2)

Ordinary Differential Equations

453

(100)

First-Order Differential Equations

453

(19)

General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems

453

(3)

Equations Solved for the Derivative. Simplest Techniques of Integration

456

(2)

Exact Differential Equations. Integrating Factor

458

(2)

Riccati Equation

460

(2)

Abel Equations of the First Kind

462

(2)

Abel Equations of the Second Kind

464

(1)

Equations Not Solved for the Derivative

465

(3)

Contact Transformations

468

(1)

Approximate Analytic Methods for Solution of Equations

469

(2)

Numerical Integration of Differential Equations

471

(1)

Second-Order Linear Differential Equations

472

(16)

Formulas for the General Solution. Some Transformations

472

(3)

Representation of Solutions as a Series in the Independent Variable

475

(2)

Asymptotic Solutions

477

(3)

Boundary Value Problems

480

(2)

Eigenvalue Problems

482

(5)

Theorems on Estimates and Zeros of Solutions

487

(1)

Second-Order Nonlinear Differential Equations

488

(26)

Form of the General Solution. Cauchy Problem

488

(1)

Equations Admitting Reduction of Order

489

(3)

Methods of Regular Series Expansions with Respect to the Independent Variable

492

(2)

Movable Singularities of Solutions of Ordinary Differential Equations. Painleve Transcendents

494

(5)

Perturbation Methods of Mechanics and Physics

499

(9)

Galerkin Method and Its Modifications (Projection Methods)

508

(3)

Iteration and Numerical Methods

511

(3)

Linear Equations of Arbitrary Order

514

(10)

Linear Equations with Constant Coefficients

514

(4)

Linear Equations with Variable Coefficients

518

(4)

Asymptotic Solutions of Linear Equations

522

(1)

Collocation Method and Its Convergence

523

(1)

Nonlinear Equations of Arbitrary Order

524

(4)

Structure of the General Solution. Cauchy Problem

524

(1)

Equations Admitting Reduction of Order

525

(3)

Linear Systems of Ordinary Differential Equations

528

(14)

Systems of Linear Constant-Coefficient Equations

528

(11)

Systems of Linear Variable-Coefficient Equations

539

(3)

Nonlinear Systems of Ordinary Differential Equations

542

(11)

Solutions and First Integrals. Uniqueness and Existence Theorems

542

(3)

Integrable Combinations. Autonomous Systems of Equations

545

(1)

Elements of Stability Theory

546

(4)

References for Chapter 12

550

(3)

First-Order Partial Differential Equations

553

(32)

Linear and Quasilinear Equations

553

(17)

Characteristic System. General Solution

553

(3)

Cauchy Problem. Existence and Uniqueness Theorem

556

(2)

Qualitative Features and Discontinuous Solutions of Quasilinear Equations

558

(9)

Quasilinear Equations of General Form. Generalized Solution, Jump Condition, and Stability Condition

567

(3)

Nonlinear Equations

570

(15)

Solution Methods

570

(6)

Cauchy Problem. Existence and Uniqueness Theorem

576

(3)

Generalized Viscosity Solutions and Their Applications

579

(5)

References for Chapter 13

584

(1)

Linear Partial Differential Equations

585

(68)

Classification of Second-Order Partial Differential Equations

585

(5)

Equations with Two Independent Variables

585

(4)

Equations with Many Independent Variables

589

(1)

Basic Problems of Mathematical Physics

590

(4)

Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems

590

(3)

First, Second, Third, and Mixed Boundary Value Problems

593

(1)

Properties and Exact Solutions of Linear Equations

594

(8)

Homogeneous Linear Equations and Their Particular Solutions

594

(4)

Nonhomogeneous Linear Equations and Their Particular Solutions

598

(2)

General Solutions of Some Hyperbolic Equations

600

(2)

Method of Separation of Variables (Fourier Method)

602

(9)

Description of the Method of Separation of Variables. General Stage of Solution

602

(3)

Problems for Parabolic Equations: Final Stage of Solution

605

(2)

Problems for Hyperbolic Equations: Final Stage of Solution

607

(2)

Solution of Boundary Value Problems for Elliptic Equations

609

(2)

Integral Transforms Method

611

(4)

Laplace Transform and Its Application in Mathematical Physics

611

(3)

Fourier Transform and Its Application in Mathematical Physics

614

(1)

Representation of the Solution of the Cauchy Problem via the Fundamental Solution

615

(3)

Cauchy Problem for Parabolic Equations

615

(2)

Cauchy Problem for Hyperbolic Equations

617

(1)

Boundary Value Problems for Parabolic Equations with One Space Variable. Green's Function

618

(5)

Representation of Solutions via the Green's Function

618

(2)

Problems for Equation

620

(3)

Boundary Value Problems for Hyperbolic Equations with One Space Variable. Green's Function. Goursat Problem

623

(8)

Representation of Solutions via the Green's Function

623

(1)

Problems for Equation

624

(2)

Problems for Equation

626

(1)

Generalized Cauchy Problem with Initial Conditions Set Along a Curve

627

(2)

Goursat Problem (a Problem with Initial Data of Characteristics)

629

(2)

Boundary Value Problems for Elliptic Equations with Two Space Variables

631

(3)

Problems and the Green's Functions for Equation

631

(2)

Representation of Solutions to Boundary Value Problems via the Green's Functions

633

(1)

Boundary Value Problems with Many Space Variables. Representation of Solutions via the Green's Function

634

(5)

Problems for Parabolic Equations

634

(2)

Problems for Hyperbolic Equations

636

(1)

Problems for Elliptic Equations

637

(1)

Comparison of the Solution Structures for Boundary Value Problems for Equations of Various Types

638

(1)

Construction of the Green's Functions. General Formulas and Relations

639

(7)

Green's Functions of Boundary Value Problems for Equations of Various Types in Bounded Domains

639

(1)

Green's Functions Admitting Incomplete Separation of Variables

640

(2)

Construction of Green's Functions via Fundamental Solutions

642

(4)

Duhamel's Principles in Nonstationary Problems

646

(3)

Problems for Homogeneous Linear Equations

646

(2)

Problems for Nonhomogeneous Linear Equations

648

(1)

Transformations Simplifying Initial and Boundary Conditions

649

(4)

Transformations That Lead to Homogeneous Boundary Conditions

649

(1)

Transformations That Lead to Homogeneous Initial and Boundary Conditions

650

(1)

References for Chapter 14

650

(3)

Nonlinear Partial Differential Equations

653

(148)

Classification of Second-Order Nonlinear Equations

653

(2)

Classification of Semilinear Equations in Two Independent Variables

653

(1)

Classification of Nonlinear Equations in Two Independent Variables

653

(2)

Transformations of Equations of Mathematical Physics

655

(12)

Point Transformations: Overview and Examples

655

(2)

Hodograph Transformations (Special Point Transformations)

657

(3)

Contact Transformations. Legendre and Euler Transformations

660

(3)

Backlund Transformations. Differential Substitutions

663

(3)

Differential Substitutions

666

(1)

Traveling-Wave Solutions, Self-Similar Solutions, and Some Other Simple Solutions. Similarity Method

667

(11)

Preliminary Remarks

667

(1)

Traveling-Wave Solutions. Invariance of Equations Under Translations

667

(2)

Self-Similar Solutions. Invariance of Equations Under Scaling Transformations

669

(5)

Equations Invariant Under Combinations of Translation and Scaling Transformations, and Their Solutions

674

(3)

Generalized Self-Similar Solutions

677

(1)

Exact Solutions with Simple Separation of Variables

678

(3)

Multiplicative and Additive Separable Solutions

678

(1)

Simple Separation of Variables in Nonlinear Partial Differential Equations

678

(1)

Complex Separation of Variables in Nonlinear Partial Differential Equations

679

(2)

Method of Generalized Separation of Variables

681

(16)

Structure of Generalized Separable Solutions

681

(2)

Simplified Scheme for Constructing Solutions Based on Presetting One System of Coordinate Functions

683

(1)

Solution of Functional Differential Equations by Differentiation

684

(4)

Solution of Functional-Differential Equations by Splitting

688

(5)

Titov--Galaktionov Method

693

(4)

Method of Functional Separation of Variables

697

(11)

Structure of Functional Separable Solutions. Solution by Reduction to Equations with Quadratic Nonlinearities

697

(1)

Special Functional Separable Solutions. Generalized Traveling-Wave Solutions

697

(3)

Differentiation Method

700

(4)

Splitting Method. Solutions of Some Nonlinear Functional Equations and Their Applications

704

(4)

Direct Method of Symmetry Reductions of Nonlinear Equations

708

(8)

Clarkson--Kruskal Direct Method

708

(4)

Some Modifications and Generalizations

712

(4)

Classical Method of Studying Symmetries of Differential Equations

716

(16)

One-Parameter Transformations and Their Local Properties

716

(3)

Symmetries of Nonlinear Second-Order Equations. Invariance Condition

719

(5)

Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions

724

(6)

Some Generalizations. Higher-Order Equations

730

(2)

Nonclassical Method of Symmetry Reductions

732

(5)

Description of the Method. Invariant Surface Condition

732

(1)

Examples: The Newell--Whitehead Equation and a Nonlinear Wave Equation

733

(4)

Differential Constraints Method

737

(11)

Description of the Method

737

(2)

First-Order Differential Constraints

739

(5)

Second- and Higher-Order Differential Constraints

744

(2)

Connection Between the Differential Constraints Method and Other Methods

746

(2)

Painleve Test for Nonlinear Equations of Mathematical Physics

748

(7)

Solutions of Partial Differential Equations with a Movable Pole. Method Description

748

(2)

Examples of Performing the Painleve Test and Truncated Expansions for Studying Nonlinear Equations

750

(3)

Construction of Solutions of Nonlinear Equations That Fail the Painleve Test, Using Truncated Expansions

753

(2)

Methods of the Inverse Scattering Problem (Soliton Theory)

755

(11)

Method Based on Using Lax Pairs

755

(2)

Method Based on a Compatibility Condition for Systems of Linear Equations

757

(3)

Solution of the Cauchy Problem by the Inverse Scattering Problem Method

760

(6)

Conservation Laws and Integrals of Motion

766

(4)

Basic Definitions and Examples

766

(1)

Equations Admitting Variational Formulation. Noetherian Symmetries

767

(3)

Nonlinear Systems of Partial Differential Equations

770

(31)

Overdetermined Systems of Two Equations

770

(2)

Pfaffian Equations and Their Solutions. Connection with Overdetermined Systems

772

(3)

Systems of First-Order Equations Describing Convective Mass Transfer with Volume Reaction

775

(5)

First-Order Hyperbolic Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type

780

(16)

Systems of Second-Order Equations of Reaction-Diffusion Type

796

(2)

References for Chapter 15

798

(3)

Integral Equations

801

(72)

Linear Integral Equations of the First Kind with Variable Integration Limit

801

(9)

Volterra Equations of the First Kind

801

(1)

Equations with Degenerate Kernel:

802

(2)

Equations with Difference Kernel: K(x, t) = K(x -- t)

804

(3)

Reduction of Volterra Equations of the First Kind to Volterra Equations of the Second Kind

807

(1)

Method of Quadratures

808

(2)

Linear Integral Equations of the Second Kind with Variable Integration Limit

810

(14)

Volterra Equations of the Second Kind

810

(1)

Equations with Degenerate Kernel:

811

(2)

Equations with Difference Kernel: K(x, t) = K(x -- t)

813

(2)

Construction of Solutions of Integral Equations with Special Right-Hand Side

815

(3)

Method of Model Solutions

818

(4)

Successive Approximation Method

822

(1)

Method of Quadratures

823

(1)

Linear Integral Equations of the First Kind with Constant Limits of Integration

824

(5)

Fredholm Integral Equations of the First Kind

824

(1)

Method of Integral Transforms

825

(2)

Regularization Methods

827

(2)

Linear Integral Equations of the Second Kind with Constant Limits of Integration

829

(27)

Fredholm Integral Equations of the Second Kind. Resolvent

829

(1)

Fredholm Equations of the Second Kind with Degenerate Kernel

830

(2)

Solution as a Power Series in the Parameter. Method of Successive Approximations

832

(2)

Fredholm Theorems and the Fredholm Alternative

834

(1)

Fredholm Integral Equations of the Second Kind with Symmetric Kernel

835

(6)

Methods of Integral Transforms

841

(3)

Method of Approximating a Kernel by a Degenerate One

844

(3)

Collocation Method

847

(2)

Method of Least Squares

849

(1)

Bubnov--Galerkin Method

850

(2)

Quadrature Method

852

(2)

Systems of Fredholm Integral Equations of the Second Kind

854

(2)

Nonlinear Integral Equations

856

(17)

Nonlinear Volterra and Urysohn Integral Equations

856

(1)

Nonlinear Volterra Integral Equations

856

(7)

Equations with Constant Integration Limits

863

(8)

References for Chapter 16

871

(2)

Difference Equations and Other Functional Equations

873

(64)

Difference Equations of Integer Argument

873

(12)

First-Order Linear Difference Equations of Integer Argument

873

(1)

First-Order Nonlinear Difference Equations of Integer Argument

874

(3)

Second-Order Linear Difference Equations with Constant Coefficients

877

(2)

Second-Order Linear Difference Equations with Variable Coefficients

879

(2)

Linear Difference Equations of Arbitrary Order with Constant Coefficients

881

(1)

Linear Difference Equations of Arbitrary Order with Variable Coefficients

882

(2)

Nonlinear Difference Equations of Arbitrary Order

884

(1)

Linear Difference Equations with a Single Continuous Variable

885

(22)

First-Order Linear Difference Equations

885

(9)

Second-Order Linear Difference Equations with Integer Differences

894

(4)

Linear mth-Order Difference Equations with Integer Differences

898

(6)

Linear mth-Order Difference Equations with Arbitrary Differences

904

(3)

Linear Functional Equations

907

(11)

Iterations of Functions and Their Properties

907

(3)

Linear Homogeneous Functional Equations

910

(2)

Linear Nonhomogeneous Functional Equations

912

(4)

Linear Functional Equations Reducible to Linear Difference Equations with Constant Coefficients

916

(2)

Nonlinear Difference and Functional Equations with a Single Variable

918

(4)

Nonlinear Difference Equations with a Single Variable

918

(1)

Reciprocal (Cyclic) Functional Equations

919

(2)

Nonlinear Functional Equations Reducible to Difference Equations

921

(1)

Power Series Solution of Nonlinear Functional Equations

922

(1)

Functional Equations with Several Variables

922

(15)

Method of Differentiation in a Parameter

922

(3)

Method of Differentiation in Independent Variables

925

(1)

Method of Substituting Particular Values of Independent Arguments

926

(2)

Method of Argument Elimination by Test Functions

928

(2)

Bilinear Functional Equations and Nonlinear Functional Equations Reducible to Bilinear Equations

930

(5)

References for Chapter 17

935

(2)

Special Functions and Their Properties

937

(54)

Some Coefficients, Symbols, and Numbers

937

(2)

Binomial Coefficients

937

(1)

Pochhammer Symbol

938

(1)

Bernoulli Numbers

938

(1)

Euler Numbers

939

(1)

Error Functions. Exponential and Logarithmic Integrals

939

(2)

Error Function and Complementary Error Function

939

(1)

Exponential Integral

940

(1)

Logarithmic Integral

941

(1)

Sine Integral and Cosine Integral. Fresnel Integrals

941

(2)

Sine Integral

941

(1)

Cosine Integral

942

(1)

Fresnel Integrals

942

(1)

Gamma Function, Psi Function, and Beta Function

943

(3)

Gamma Function

943

(1)

Psi Function (Digamma Function)

944

(1)

Beta Function

945

(1)

Incomplete Gamma and Beta Functions

946

(1)

Incomplete Gamma Function

946

(1)

Incomplete Beta Function

947

(1)

Bessel Functions (Cylindrical Functions)

947

(6)

Definitions and Basic Formulas

947

(2)

Integral Representations and Asymptotic Expansions

949

(2)

Zeros and Orthogonality Properties of Bessel Functions

951

(1)

Hankel Functions (Bessel Functions of the Third Kind)

952

(1)

Modified Bessel Functions

953

(2)

Definitions. Basic Formulas

953

(1)

Integral Representations and Asymptotic Expansions

954

(1)

Airy Functions

955

(1)

Definition and Basic Formulas

955

(1)

Power Series and Asymptotic Expansions

956

(1)

Degenerate Hypergeometric Functions (Kummer Functions)

956

(4)

Definitions and Basic Formulas

956

(3)

Integral Representations and Asymptotic Expansions

959

(1)

Whittaker Functions

960

(1)

Hypergeometric Functions

960

(2)

Various Representations of the Hypergeometric Function

960

(1)

Basic Properties

960

(2)

Legendre Polynomials, Legendre Functions, and Associated Legendre Functions

962

(5)

Legendre Polynomials and Legendre Functions

962

(2)

Associated Legendre Functions with Integer Indices and Real Argument

964

(1)

Associated Legendre Functions. General Case

965

(2)

Parabolic Cylinder Functions

967

(2)

Definitions. Basic Formulas

967

(1)

Integral Representations, Asymptotic Expansions, and Linear Relations

968

(1)

Elliptic Integrals

969

(3)

Complete Elliptic Integrals

969

(1)

Incomplete Elliptic Integrals (Elliptic Integrals)

970

(2)

Elliptic Functions

972

(6)

Jacobi Elliptic Functions

972

(4)

Weierstrass Elliptic Function

976

(2)

Jacobi Theta Functions

978

(2)

Series Representation of the Jacobi Theta Functions. Simplest Properties

978

(1)

Various Relations and Formulas. Connection with Jacobi Elliptic Functions

978

(2)

Mathieu Functions and Modified Mathieu Functions

980

(2)

Mathieu Functions

980

(2)

Modified Mathieu Functions

982

(1)

Orthogonal Polynomials

982

(6)

Laguerre Polynomials and Generalized Laguerre Polynomials

982

(1)

Chebyshev Polynomials and Functions

983

(2)

Hermite Polynomials

985

(1)

Jacobi Polynomials and Gegenbauer Polynomials

986

(2)

Nonorthogonal Polynomials

988

(3)

Bernoulli Polynomials

988

(1)

Euler Polynomials

989

(1)

References for Chapter 18

990

(1)

Calculus of Variations and Optimization

991

(40)

Calculus of Variations and Optimal Control

991

(21)

Some Definitions and Formulas

991

(2)

Simplest Problem of Calculus of Variations

993

(9)

Isoperimetric Problem

1002

(4)

Problems with Higher Derivatives

1006

(2)

Lagrange Problem

1008

(2)

Pontryagin Maximum Principle

1010

(2)

Mathematical Programming

1012

(19)

Linear Programming

1012

(15)

Nonlinear Programming

1027

(1)

References for Chapter 19

1028

(3)

Probability Theory

1031

(50)

Simplest Probabilistic Models

1031

(8)

Probabilities of Random Events

1031

(4)

Conditional Probability and Simplest Formulas

1035

(2)

Sequences of Trials

1037

(2)

Random Variables and Their Characteristics

1039

(29)

One-Dimensional Random Variables

1039

(3)

Characteristics of One-Dimensional Random Variables

1042

(5)

Main Discrete Distributions

1047

(4)

Continuous Distributions

1051

(6)

Multivariate Random Variables

1057

(11)

Limit Theorems

1068

(3)

Convergence of Random Variables

1068

(1)

Limit Theorems

1069

(2)

Stochastic Processes

1071

(10)

Theory of Stochastic Processes

1071

(3)

Models of Stochastic Processes

1074

(5)

References for Chapter 20

1079

(2)

Mathematical Statistics

1081

(30)

Introduction to Mathematical Statistics

1081

(7)

Basic Notions and Problems of Mathematical Statistics

1081

(1)

Simplest Statistical Transformations

1082

(5)

Numerical Characteristics of Statistical Distribution

1087

(1)

Statistical Estimation

1088

(6)

Estimators and Their Properties

1088

(3)

Estimation Methods for Unknown Parameters

1091

(2)

Interval Estimators (Confidence Intervals)

1093

(1)

Statistical Hypothesis Testing

1094

(17)

Statistical Hypothesis. Test

1094

(4)

Goodness-of-Fit Tests

1098

(3)

Problems Related to Normal Samples

1101

(8)

References for Chapter 21

1109

(2)

Part II. Mathematical Tables

1111

(340)

Finite Sums and Infinite Series

1113

(16)

Finite Sums

1113

(5)

Numerical Sum

1113

(3)

Functional Sums

1116

(2)

Infinite Series

1118

(11)

Numerical Series

1118

(2)

Functional Series

1120

(7)

References for Chapter T1

1127

(2)

Integrals

1129

(28)

Indefinite Integrals

1129

(18)

Integrals Involving Rational Functions

1129

(5)

Integrals Involving Irrational Functions

1134

(3)

Integrals Involving Exponential Functions

1137

(1)

Integrals Involving Hyperbolic Functions

1137

(3)

Integrals Involving Logarithmic Functions

1140

(2)

Integrals Involving Trigonometric Functions

1142

(5)

Integrals Involving Inverse Trigonometric Functions

1147

(10)

Tables of Definite Integrals

1147

(1)

Integrals Involving Power-Law Functions

1147

(3)

Integrals Involving Exponential Functions

1150

(2)

Integrals Involving Hyperbolic Functions

1152

(1)

Integrals Involving Logarithmic Functions

1152

(1)

Integrals Involving Trigonometric Functions

1153

(2)

References for Chapter T2

1155

(2)

Integral Transforms

1157

(38)

Tables of Laplace Transforms

1157

(7)

General Formulas

1157

(2)

Expressions with Power-Law Functions

1159

(1)

Expressions with Exponential Functions

1159

(1)

Expressions with Hyperbolic Functions

1160

(1)

Expressions with Logarithmic Functions

1161

(1)

Expressions with Trigonometric Functions

1161

(2)

Expressions with Special Functions

1163

(1)

Tables of Inverse Laplace Transforms

1164

(13)

General Formulas

1164

(2)

Expressions with Rational Functions

1166

(4)

Expressions with Square Roots

1170

(2)

Expressions with Arbitrary Powers

1172

(1)

Expressions with Exponential Functions

1172

(2)

Expressions with Hyperbolic Functions

1174

(1)

Expressions with Logarithmic Functions

1174

(1)

Expressions with Trigonometric Functions

1175

(1)

Expressions with Special Functions

1176

(1)

Tables of Fourier Cosine Transforms

1177

(5)

General Formulas

1177

(1)

Expressions with Power-Law Functions

1177

(1)

Expressions with Exponential Functions

1178

(1)

Expressions with Hyperbolic Functions

1179

(1)

Expressions with Logarithmic Functions

1179

(1)

Expressions with Trigonometric Functions

1180

(1)

Expressions with Special Functions

1181

(1)

Tables of Fourier Sine Transforms

1182

(5)

General Formulas

1182

(1)

Expressions with Power-Law Functions

1182

(1)

Expressions with Exponential Functions

1183

(1)

Expressions with Hyperbolic Functions

1184

(1)

Expressions with Logarithmic Functions

1184

(1)

Expressions with Trigonometric Functions

1185

(1)

Expressions with Special Functions

1186

(1)

Tables of Mellin Transforms

1187

(3)

General Formulas

1187

(1)

Expressions with Power-Law Functions

1188

(1)

Expressions with Exponential Functions

1188

(1)

Expressions with Logarithmic Functions

1189

(1)

Expressions with Trigonometric Functions

1189

(1)

Expressions with Special Functions

1190

(1)

Tables of Inverse Mellin Transforms

1190

(5)

Expressions with Power-Law Functions

1190

(1)

Expressions with Exponential and Logarithmic Functions

1191

(1)

Expressions with Trigonometric Functions

1192

(1)

Expressions with Special Functions

1193

(1)

References for Chapter T3

1194

(1)

Orthogonal Curvilinear Systems of Coordinate

1195

(12)

Arbitrary Curvilinear Coordinate Systems

1195

(3)

General Nonorthogonal Curvilinear Coordinates

1195

(1)

General Orthogonal Curvilinear Coordinates

1196

(2)

Special Curvilinear Coordinate Systems

1198

(9)

Cylindrical Coordinates

1198

(1)

Spherical Coordinates

1199

(1)

Coordinates of a Prolate Ellipsoid of Revolution

1200

(1)

Coordinates of an Oblate Ellipsoid of Revolution

1201

(1)

Coordinates of an Elliptic Cylinder

1202

(1)

Conical Coordinates

1202

(1)

Parabolic Cylinder Coordinates

1203

(1)

Parabolic Coordinates

1203

(1)

Bicylindrical Coordinates

1204

(1)

Bipolar Coordinates (in Space)

1204

(1)

Toroidal Coordinates

1205

(1)

References for Chapter T4

1205

(2)

Ordinary Differential Equations

1207

(22)

First-Order Equations

1207

(5)

Second-Order Linear Equations

1212

(11)

Equations Involving Power Functions

1213

(7)

Equations Involving Exponential and Other Functions

1220

(2)

Equations Involving Arbitrary Functions

1222

(1)

Second-Order Nonlinear Equations

1223

(6)

Equations of the Form

1223

(2)

Equations of the Form

1225

(3)

References for Chapter T5

1228

(1)

Systems of Ordinary Differential Equations

1229

(18)

Linear Systems of Two Equations

1229

(8)

Systems of First-Order Equations

1229

(3)

Systems of Second-Order Equations

1232

(5)

Linear Systems of Three and More Equations

1237

(2)

Nonlinear Systems of Two Equations

1239

(5)

Systems of First-Order Equations

1239

(1)

Systems of Second-Order Equations

1240

(4)

Nonlinear Systems of Three or More Equations

1244

(3)

References for Chapter T6

1246

(1)

First-Order Partial Differential Equations

1247

(20)

Linear Equations

1247

(5)

Equations of the Form

1247

(1)

Equations of the Form

1248

(2)

Equations of the Form

1250

(2)

Quasilinear Equations

1252

(6)

Equations of the Form

1252

(2)

Equations of the Form

1254

(2)

Equations of the Form

1256

(2)

Nonlinear Equations

1258

(9)

Equations Quadratic in One Derivative

1258

(1)

Equations Quadratic in Two Derivatives

1259

(2)

Equations with Arbitrary Nonlinearities in Derivatives

1261

(4)

References for Chapter T7

1265

(2)

Linear Equations and Problems of Mathematical Physics

1267

(34)

Parabolic Equations

1267

(11)

Heat Equation

1267

(1)

Nonhomogeneous Heat Equation

1268

(2)

Equation of the Form

1270

(1)

Heat Equation with Axial Symmetry

1270

(1)

Equation of the Form

1271

(1)

Heat Equation with Central Symmetry

1272

(1)

Equation of the Form

1273

(1)

Equation of the Form

1274

(2)

Equations of the Diffusion (Thermal) Boundary Layer

1276

(1)

Schrodinger Equation

1276

(2)

Hyperbolic Equations

1278

(6)

Wave Equation

1278

(1)

Equation of the Form

1279

(1)

Klein--Gordon Equation

1280

(1)

Equation of the Form

1281

(1)

Equation of the Form

1282

(1)

Equation of the Form

1283

(1)

Equations of the Form

1284

(1)

Elliptic Equations

1284

(10)

Laplace Equation Δw = 0

1284

(3)

Poisson Equation Δw + Φ(x) = 0

1287

(2)

Helmholtz Equation Δw + λw = --Φ(x)

1289

(5)

Fourth-Order Linear Equations

1294

(7)

Equation of the Form

1294

(1)

Equation of the Form

1295

(2)

Biharmonic Equation ΔΔw = 0

1297

(1)

Nonhomogeneous Biharmonic Equation ΔΔw = Φ(x, y)

1298

(1)

References for Chapter T8

1299

(2)

Nonlinear Mathematical Physics Equations

1301

(36)

Parabolic Equations

1301

(11)

Nonlinear Heat Equations of the Form

1301

(2)

Equations of the Form

1303

(4)

Burgers Equation and Nonlinear Heat Equation in Radial Symmetric Cases

1307

(2)

Nonlinear Schrodinger Equations

1309

(3)

Hyperbolic Equations

1312

(6)

Nonlinear Wave Equations of the Form

1312

(4)

Other Nonlinear Wave Equations

1316

(2)

Elliptic Equations

1318

(6)

Nonlinear Heat Equations of the Form

1318

(3)

Equations of the Form

1321

(1)

Equations of the Form

1322

(2)

Other Second-Order Equations

1324

(3)

Equations of Transonic Gas Flow

1324

(2)

Monge--Ampere Equations

1326

(1)

Higher-Order Equations

1327

(10)

Third-Order Equations

1327

(5)

Fourth-Order Equations

1332

(3)

References for Chapter T9

1335

(2)

Systems of Partial Differential Equations

1337

(48)

Nonlinear Systems of Two First-Order Equations

1337

(4)

Linear Systems of Two Second-Order Equations

1341

(2)

Nonlinear Systems of Two Second-Order Equations

1343

(31)

Systems of the Form

1343

(14)

Systems of the Form

1357

(7)

Systems of the Form

1364

(4)

Systems of the Form

1368

(5)

Other Systems

1373

(1)

Systems of General Form

1374

(11)

Linear Systems

1374

(1)

Nonlinear Systems of Two Equations Involving the First Derivatives in t

1374

(4)

Nonlinear Systems of Two Equations Involving the Second Derivatives in t

1378

(3)

Nonlinear Systems of Many Equations Involving the First Derivatives in t

1381

(1)

References for Chapter T10

1382

(3)

Integral Equations

1385

(24)

Linear Equations of the First Kind with Variable Limit of Integration

1385

(6)

Linear Equations of the Second Kind with Variable Limit of Integration

1391

(5)

Linear Equations of the First Kind with Constant Limits of Integration

1396

(5)

Linear Equations of the Second Kind with Constant Limits of Integration

1401

(8)

References for Chapter T11

1406

(3)

Functional Equations

1409

(42)

Linear Functional Equations in One Independent Variable

1409

(19)

Linear Difference and Functional Equations Involving Unknown Function with Two Different Arguments

1409

(12)

Other Linear Functional Equations

1421

(7)

Nonlinear Functional Equations in One Independent Variable

1428

(10)

Functional Equations with Quadratic Nonlinearity

1428

(5)

Functional Equations with Power Nonlinearity

1433

(1)

Nonlinear Functional Equation of General Form

1434

(4)

Functional Equations in Several Independent Variables

1438

(13)

Linear Functional Equations

1438

(5)

Nonlinear Functional Equations

1443

(7)

References for Chapter T12

1450

(1)

Supplement. Some Useful Electronic Mathematical Resources

1451

(2)

Index

1453

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