Advanced mathematical methods in science and engineering / S.I. Hayek.

Format:

Book

Author/Creator:

Hayek, Sabih I., 1938-

Contributor:

Rosengarten Family Fund.

Language:

English

Subjects (All):

Mathematics.

Physical Description:

xv, 734 pages : illustrations ; 26 cm

Place of Publication:

New York : M. Dekker, [2001]

Contents:

1 Ordinary Differential Equations 1

1.2 Linear Differential Equations of First Order 2

1.3 Linear Independence and the Wronskian 3

1.4 Linear Homogeneous Differential Equation of Order n with Constant Coefficients 4

1.5 Euler's Equation 6

1.6 Particular Solutions by Method of Undetermined Coefficients 7

1.7 Particular Solutions by the Method of Variations of Parameters 9

1.8 Abel's Formula for the Wronskian 12

1.9 Initial Value Problems 13

2 Series Solutions of Ordinary Differential Equations 19

2.2 Power Series Solutions 20

2.3 Classification of Singularities 23

2.4 Frobenius Solution 25

3 Special Functions 43

3.1 Bessel Functions 43

3.2 Bessel Function of Order Zero 45

3.3 Bessel Function of an Integer Order n 47

3.4 Recurrence Relations for Bessel Functions 49

3.5 Bessel Functions of Half Orders 51

3.6 Spherical Bessel Functions 52

3.7 Hankel Functions 53

3.8 Modified Bessel Functions 54

3.9 Generalized Equations Leading to Solutions in Terms of Bessel Functions 56

3.10 Bessel Coefficients 58

3.11 Integral Representation of Bessel Functions 62

3.12 Asymptotic Approximations of Bessel Functions for Small Arguments 65

3.13 Asymptotic Approximations of Bessel Functions for Large Arguments 66

3.14 Integrals of Bessel Functions 66

3.15 Zeroes of Bessel Functions 68

3.16 Legendre Functions 69

3.17 Legendre Coefficients 75

3.18 Recurrence Formulae for Legendre Polynomials 77

3.19 Integral Representation for Legendre Polynomials 79

3.20 Integrals of Legendre Polynomials 81

3.21 Expansions of Functions in Terms of Legendre Polynomials 85

3.22 Legendre Function of the Second Kind Q[subscript n](x) 89

3.23 Associated Legendre Functions 93

3.24 Generating Function for Associated Legendre Functions 94

3.25 Recurrence Formulae for P[superscript m subscript n] 95

3.26 Integrals of Associated Legendre Functions 96

3.27 Associated Legendre Function of the Second Kind Q[superscript m subscript n] 97

4 Boundary Value Problems and Eigenvalue Problems 107

4.2 Vibration, Wave Propagation or Whirling of Stretched Strings 109

4.3 Longitudinal Vibration and Wave Propagation in Elastic Bars 113

4.4 Vibration, Wave Propagation and Whirling of Beams 117

4.5 Waves in Acoustic Horns 124

4.6 Stability of Compressed Columns 127

4.7 Ideal Transmission Lines (Telegraph Equation) 130

4.8 Torsional Vibration of Circular Bars 132

4.9 Orthogonality and Orthogonal Sets of Functions 133

4.10 Generalized Fourier Series 135

4.11 Adjoint Systems 138

4.12 Boundary Value Problems 140

4.13 Eigenvalue Problems 142

4.14 Properties of Eigenfunctions of Self-Adjoint Systems 144

4.15 Sturm-Liouville System 148

4.16 Sturm-Liouville System for Fourth Order Equations 155

4.17 Solution of Non-Homogeneous Eigenvalue Problems 158

4.18 Fourier Sine Series 161

4.19 Fourier Cosine Series 163

4.20 Complete Fourier Series 165

4.21 Fourier-Bessel Series 169

4.22 Fourier-Legendre Series 171

5 Functions of A Complex Variable 185

5.1 Complex Numbers 185

5.2 Analytic Functions 189

5.3 Elementary Functions 201

5.4 Integration in the Complex Plane 207

5.5 Cauchy's Integral Theorem 210

5.6 Cauchy's Integral Formula 213

5.7 Infinite Series 216

5.8 Taylor's Expansion Theorem 217

5.9 Laurent's Series 222

5.10 Classification of Singularities 229

5.11 Residues and Residue Theorem 231

5.12 Integrals of Periodic Functions 236

5.13 Improper Real Integrals 237

5.14 Improper Real Integrals Involving Circular Functions 239

5.15 Improper Real Integrals of Functions Having Singularities on the Real Axis 242

5.16 Theorems on Limiting Contours 245

5.17 Evaluation of Real Improper Integrals By Non-Circular Contours 249

5.18 Integrals of Even Functions Involving log x 252

5.19 Integrals of Functions Involving x[superscript a] 259

5.20 Integrals of Odd or Asymmetric Functions 263

5.21 Integrals of Odd or Asymmetric Functions Involving log x 264

5.22 Inverse Laplace Transforms 266

6 Partial Differential Equations of Mathematical Physics 293

6.2 The Diffusion Equation 293

6.3 The Vibration Equation 297

6.4 The Wave Equation 302

6.5 Helmholtz Equation 307

6.6 Poisson and Laplace Equations 308

6.7 Classification of Partial Differential Equations 312

6.8 Uniqueness of Solutions 312

6.9 The Laplace Equation 319

6.10 The Poisson Equation 332

6.11 The Helmholtz Equation 336

6.12 The Diffusion Equation 342

6.13 The Vibration Equation 349

6.14 The Wave Equation 355

7 Integral Transforms 383

7.1 Fourier Integral Theorem 383

7.2 Fourier Cosine Transform 384

7.3 Fourier Sine Transform 385

7.4 Complex Fourier Transform 385

7.5 Multiple Fourier Transform 386

7.6 Hankel Transform of Order Zero 387

7.7 Hankel Transform of Order v 389

7.8 General Remarks About Transforms Derived from the Fourier Integral Theorem 393

7.9 Generalized Fourier Transform 393

7.10 Two-Sided Laplace Transform 399

7.11 One-Sided Generalized Fourier Transform 399

7.12 Laplace Transform 400

7.13 Mellin Transform 401

7.14 Operational Calculus with Laplace Transforms 402

7.15 Solution of Ordinary and Partial Differential Equations by Laplace Transforms 411

7.16 Operational Calculus with Fourier Consine Transform 421

7.17 Operational Calculus with Fourier Sine Transform 425

7.18 Operational Calculus with Complex Fourier Transform 431

7.19 Operational Calculus with Multiple Fourier Transform 435

7.20 Operational Calculus with Hankel Transform 438

8 Green's Functions 453

8.2 Green's Function for Ordinary Differential Boundary Value Problems 453

8.3 Green's Function for an Adjoint System 455

8.4 Symmetry of the Green's Functions and Reciprocity 456

8.5 Green's Function for Equations with Constant Coefficients 458

8.6 Green's Functions for Higher Ordered Sources 459

8.7 Green's Function for Eigenvalue Problems 459

8.8 Green's Function for Semi-Infinite One-Dimensional Media 462

8.9 Green's Function for Infinite One-Dimensional Media 465

8.10 Green's Function for Partial Differential Equations 466

8.11 Green's Identities for the Laplacian Operator 468

8.12 Green's IDentity for the Helmholtz Operator 469

8.13 Green's Identity for Bi-Laplacian Operator 469

8.14 Green's Identity for the Diffusion Operator 470

8.15 Green's Identity for the Wave Operator 471

8.16 Green's Function for Unbounded Media

Fundamental Solution 472

8.17 Fundamental Solution for the Laplacian 473

8.18 Fundamental Solution for the Bi-Laplacian 476

8.19 Fundamental Solution for the Helmholtz Operator 477

8.20 Fundamental Solution for the Operator, - [down triangle, open superscript 2] [mu superscript 2] 479

8.21 Causal Fundamental Solution for the Diffusion Operator 480

8.22 Causal Fundamental Solution for the Wave Operator 482

8.23 Fundamental Solutions for the Bi-Laplacian Helmholtz Operator 484

8.24 Green's Function for the Laplacian Operator for Bounded Media 485

8.25 Construction of the Auxiliary Function

Method of Images 488

8.26 Green's Function for the Laplacian for Half-Space 488

8.27 Green's Function for the Laplacian by Eigenfunction Expansion for Bounded Media 492

8.28 Green's Function for a Circular Area for the Laplacian 493

8.29 Green's Function for Spherical Geometry for the Laplacian 500

8.30 Green's Function for the Helmholtz Operator for Bounded Media 503

8.31 Green's Function for the Helmholtz Operator for Half-Space 503

8.32 Green's Function for a Helmholtz Operator in Quarter-Space 507

8.33 Causal Green's Function for the Wave Operator in Bounded Media 510

8.34 Causal Green's Function for the Diffusion Operator for Bounded Media 515

8.35 Method of Summation of Series Solutions in Two-Dimensional Media 519

9 Asymptotic Methods 537

9.2 Method of

Integration by Parts 537

9.3 Laplace's Integral 538

9.4 Steepest Descent Method 539

9.5 Debye's First Order Approximation 543

9.6 Asymptotic Series Approximation 548

9.7 Method of Stationary Phase 552

9.8 Steepest Descent Method in Two Dimensions 553

9.9 Modified Saddle Point Method

Subtraction of A Simple Pole 554

9.10 Modified Saddle Point Method: Subtraction of Pole of Order N 558

9.11 Solution of Ordinary Differential Equations for Large Arguments 559

9.12 Classification of Points At Infinity 559

9.13 Solutions of Ordinary Differential Equations with Regular Singular Points 561

9.14 Asymptotic Solutions of Ordinary Differential Equations with Irregular Singular Points of Rank One 563

9.15 The Phase Integral and Wkbj Method for An Irregular Singular Point of Rank One 568

9.16 Asymptotic Solutions of Ordinary Differential Equations with Irregular Singular Points of Rank Higher Than One 571

9.17 Asymptotic Solutions of Ordinary Differential Equations with Large Parameters 574

Appendix A Infinite Series 585

A.2 Convergence Tests 586

A.3 Infinite Series of Functions of One Variable 591

A.4 Power Series 594

Appendix B Special Functions 599

B.1 The Gamma Function [Gamma](x) 599

B.2 Psi Function [psi](x) 600

B.3 Incomplete Gamma Function [gamma] (x,y) 602

B.4 Beta Function B(x,y) 603

B.5 Error Function erf(x) 604

B.6 Fresnel Functions C(x), S(x) and F(x) 606

B.7 Exponential Integrals Ei(x) and E[subscript n](x) 608

B.8 Sine and Cosine Integrals Si(x) and Ci(x) 610

B.9 Tchebyshev Polynomials T[subscript n](x) and U[subscript n](x) 612

B.10 Laguerre Polynomials L[subscript n](x) 613

B.11 Associated Laguerre Polynomials L[superscript m subscript n](x) 614

B.12 Hermite Polynomials H[subscript n](x) 615

B.13 Hypergeometric Functions F(a, b; c; x) 617

B.14 Confluent Hypergeometric Functions M(a,c,s) and U(a,c,x) 618

B.15 Kelvin Functions (ber[subscript v] (x), bei[subscript v] (x), ker[subscript v] (x), kei(x)) 620

Appendix C Orthogonal Coordinate Systems 625

C.2 Generalized Orthogonal Coordinate Systems 625

C.3 Cartesian Coordinates 627

C.4 Circular Cylindrical Coordinates 627

C.5 Elliptic-Cylindrical Coordinates 628

C.6 Spherical Coordinates 629

C.7 Prolate Spheroidal Coordinates 630

C.8 Oblate Spheroidal Coordinates 632

Appendix D Dirac Delta Functions 635

D.1 Dirac Delta Function 635

D.2 Dirac Delta Function of Order One 641

D.3 Dirac Delta Function of Order N 641

D.4 Equivalent Representations of Distributed Functions 642

D.5 Dirac Delta Functions in n-Dimensional Space 643

D.6 Spherically Symmetric Dirac Delta Function Representation 645

D.7 Dirac Delta Function of Order N in n-Dimensional Space 646

Appendix E Plots of Special Functions 651

E.1 Bessel Functions of the First and Second Kind of Order 0, 1, 2 651

E.2 Spherical Bessel Functions of the First and Second Kind of Order 0, 1, 2 652

E.3 Modified Bessel Function of the First and Second Kind of Order 0, 1, 2 653

E.4 Bessel Function of the First and Second Kind of Order 1/2 654

E.5 Modified Bessel Function of the First and Second Kind of Order 1/2 654.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

0824704665

OCLC:

44720530