Mathematics for the Physical Sciences

by Leslie Copley

Overview

The book begins with a thorough introduction to complex analysis, which is then used to understand the properties of ordinary differential equations and their solutions. The latter are obtained in both series and integral representations. Integral transforms are introduced, providing an opportunity to complement complex analysis with techniques that flow from an algebraic approach. This moves naturally into a discussion of eigenvalue and boundary vale problems. A thorough discussion of multi-dimensional boundary value problems then introduces the reader to the fundamental partial differential equations and “special functions” of mathematical physics. Moving to non-homogeneous boundary value problems the reader is presented with an analysis of Green’s functions from both analytical and algebraic points of view. This leads to a concluding chapter on integral equations.

Product Details

ISBN-13:	9783110409451
Publisher:	De Gruyter Poland
Publication date:	12/15/2014
Pages:	446
Product dimensions:	6.69(w) x 9.45(h) x 0.06(d)
Age Range:	18 Years

About the Author

Leslie Copley, Professor Emeritus of Physics, Carleton University, Canada.

Table of Contents

Foreword x

1 Functions of a Complex Variable 1

1.1 Introduction 1

1.2 Complex Numbers 1

1.2.1 Complex Arithmetic 1

1.2.2 Graphical Representation and the Polar Form 4

1.2.3 Curves and Regions in the Complex Plane 9

1.3 Functions of a Complex Variable 13

1.3.1 Basic Concepts 13

1.3.2 Continuity, Differentiability and Analyticity 14

1.4 Power Series 20

1.5 The Elementary Functions 25

1.5.1 Rational Functions 25

1.5.2 The Exponential Function 25

1.5.3 The Trigonometric and Hyperbolic Functions 26

1.5.4 The Logarithm 28

1.5.5 The General Power z^α 30

1.6 Multivalued Functions and Riemann Surfaces 30

1.7 Conformal Mapping 42

2 Cauchy's Theorem 46

2.1 Complex Integration 46

2.2 Cauchy's Theorem 52

2.2.1 Statement and Proof 52

2.2.2 Path Independence 54

2.2.3 The Fundamental Theorem of Calculus 56

2.3 Further Consequences of Cauchy's Theorem 58

2.3.1 Cauchy's Integral 58

2.3.2 Cauchy's Derivative Formula 60

2.3.3 The Maximum Modulus Principle 62

2.3.4 The Cauchy-Liouville Theorem 63

2.4 Power Series Representations of Analytic Functions 64

2.4.1 Uniform Convergence 64

2.4.2 Taylor's Theorem 67

2.4.3 Laurent's Theorem 71

2.4.4 Practical Methods for Generating Power Series 76

2.5 Zeros and Singularities 82

3 The Calculus of Residues 90

3.1 The Residue Theorem 90

3.2 Calculating Residues 92

3.3 Evaluating Definite Integrals 95

3.3.1 Angular Integrals 95

3.3.2 Improper Integrals of Rational Functions 96

3.3.3 Improper Integrals Involving Trigonometric Functions 100

3.3.4 Improper Integrals Involving Exponential Functions 104

3.3.5 Integrals Involving Many-Valued Functions 107

3.3.6 Deducing Integrals from Others 110

3.3.7 Singularities on the Contour and Principal Value Integrals 113

3.4 Summation of Series 119

3.5 Representation of Meromorphic Functions 122

4 Dispersion Representations 125

4.1 From Cauchy Integral Representation to Hilbert Transforms 125

4.2 Adding Poles and Subtractions 128

4.3 Mathematical Applications 130

5 Analytic Continuation 133

5.1 Analytic Continuation 133

5.2 The Gamma Function 141

5.3 Integral Representations and Integral Transforms 148

6 Asymptotic Expansions 150

6.1 Asymptotic Series 150

6.2 Watson's Lemma 153

6.3 The Method of Steepest Descent 155

7 Padé Approximants 163

7.1 From Power Series to Padé Sequences 163

7.2 Numerical Experiments 166

7.3 Stieltjes Functions 170

7.4 Convergence Theorems 176

7.5 Type II Padé Approximants 177

8 Fourier Series and Transforms 180

8.1 Trigonometrical and Fourier Series 180

8.2 The Convergence Question 184

8.3 Functions Having Arbitrary Period 189

8.4 Half-Range Expansions - Fourier Sine and Cosine Series 192

8.5 Complex Form of Fourier Series 194

8.6 Transition to Fourier Transforms 199

8.7 Examples of Fourier Transforms 201

8.8 The Dirac Delta Function and Transforms of Distributions 203

8.9 Properties of Fourier Transforms 214

8.10 Fourier Sine and Cosine Transforms 215

8.11 Laplace Transforms 218

8.12 Application: Solving Differential Equations 223

9 Ordinary Linear Differential Equations 234

9.1 Introduction 234

9.2 Linear DE's of Second Order 235

9.3 Given One Solution, Find the Others 238

9.4 Finding a Particular Solution for Homogeneous DE's 241

9.5 Method of Frobenius 247

9.6 The Legendre Differential Equation 251

9.7 Bessel's Differential Equation 254

9.8 Some Other Tricks of the Trade 261

9.8.1 Expansion About the Point at Infinity 261

9.8.2 Factorization of the Behaviour at Infinity 263

9.8.3 Changing the Independent Variable 264

9.8.4 Changing the Dependent Variable 265

9.9 Solution by Definite integrals 266

9.10 The Hypergeometric Equation 275

10 Partial Differential Equations and Boundary Value Problems 279

10.1 The Partial Differential Equations of Mathematical Physics 279

10.2 Curvilinear Coordinates 280

10.3 Separation of Variables 282

10.4 What a Difference the Choice of Coordinate System Makes! 287

10.5 The Sturm-Liouville Eigenvalue Problem 295

10.6 A Convenient Notation (And Another Algebraic Digression) 302

10.7 Fourier Series and Transforms as Eigenfunction Expansions 304

10.8 Normal Mode (or Initial Value) Problems 308

11 Special Functions 313

11.1 Introduction 313

11.2 Spherical Harmonics: Problems Possessing Spherical Symmetry 313

11.2.1 Introduction 313

11.2.2 Associated Legendre Polynomials 314

11.2.3 Properties of Legendre Polynomials 316

11.2.4 Problems Possessing Azimuthal Symmetry 323

11.2.5 Properties of the Associated Legendre Polynomials 326

11.2.6 Completeness and the Spherical Harmonics 329

11.2.7 Applications: Problems Without Azimuthat Symmetry 331

11.3 Bessel Functions: Problems Possessing Cylindrical Symmetry 338

11.3.1 Properties of Bessel and Neumann Functions 338

11.3.2 Applications 344

11.3.3 Modified Bessel Functions 350

11.3.4 Electrostatic Potential in and around Cylinders 352

11.3.5 Fourier-Bessel Transforms 355

11.4 Spherical Bessel Functions: Spherical Waves 358

11.4.1 Properties of Spherical Bessel Functions 358

11.4.2 Applications: Spherical Waves 362

11.5 The Classical Orthogonal Polynomials 371

11.5.1 The Polynomials and Their Properties 371

11.5.2 Applications 377

12 Non-Homogeneous Boundary Value Problems: Green's Functions 384

12.1 Ordinary Differential Equations 384

12.1.1 Definition of a Green's Function 384

12.1.2 Direct Construction of the Sturm Liouville Green's Function 385

12.1.3 Application: The Bowed Stretched String 388

12.1.4 Eigenfunction Expansions: The Bilinear Formula 390

12.1.5 Application: the Infinite Stretched String 392

12.2 Partial Differential Equations 395

12.2.1 Green's Theorem and Its Consequences 395

12.2.2 Poisson's Equation in Two Dimensions and With Rectangular Symmetry 398

12.2.3 Potential Problems in Three Dimensions and the Method of Images 401

12.2.4 Expansion of the Dirichlet Green's Function for Poisson's Equation When There Is Spherical Symmetry 403

12.2.5 Applications 405

12.3 The Non-Homogeneous Wave and Diffusion Equations 407

12.3.1 The Non-Homogeneous Helmholtz Equation 407

12.3.2 The Forced Drumhead 408

12.3.3 The Non-Homogeneous Helmholtz Equation With Boundaries at Infinity 410

12.3.4 General Time Dependence 412

12.3.5 The Wave and Diffusion Equation Green's Functions for Boundaries at Infinity 414

13 integral Equations 418

13.1 Introduction 418

13.2 Types of Integral Equations 418

13.3 Convolution Integral Equations 420

13.4 Integral Equations With Separable Kernels 422

13.5 Solution by Iteration: Neumann Series 426

13.6 Hilbert Schmidt Theory 428

Bibliography 434