Preface

1 Mathematical Preliminaries

　1.1 InfiniteSeries

　1.2 Series ofFunctions

　1.3 Binomial Theorem

　1.4 Mathematical Induction

　1.5 Operations on Series Expansions of Functions

　1.6 Some Important Series

　1.7 Vectors

　1.8 Complex Numbers and Functions

　1.9 Derivatives andExtrema

　1.10 Evaluation oflntegrals

　1.1 I Dirac Delta Function

　AdditionaIReadings

2 Determinants and Matrices

　2.1 Determinants

　2.2 Matrices

　AdditionaI Readings

3 Vector Analysis

　3.1 Review ofBasic Properties

　3.2 Vectors in 3-D Space

　3.3 Coordinate Transformations

　3.4 Rotations in IR3

　3.5 Differential Vector Operators

　3.6 Differential Vector Operators: Further Properties

　3.7 Vectorlntegration

　3.8 Integral Theorems

　3.9 PotentiaITheory

　3.10 Curvilinear Coordinates

　AdditionaIReadings

4 Tensors and Differential Forms

　4.1 TensorAnalysis

　4.2 Pseudotensors, Dual Tensors

　4.3 Tensors in General Coordinates

　4.4 Jacobians

　4.5 DifferentialForms

　4.6 DifferentiatingForms

　4.7 IntegratingForms

　AdditionalReadings

5 Vector Spaces

　5.1 Vectors in Function Spaces

　5.2 Gram-Schmidt Orthogonalization

　5.3 Operators

　5.4 SelfAdjointOperators

　5.5 Unitaty Operators

　5.6 Transformations of Operators

　5.7 Invariants

　5.8 Summary-Vector Space Notation

　AdditionaIReadings

6 Eigenvalue Problems

　6.1 EigenvalueEquations

　6.2 Matrix Eigenvalue Problems

　6.3 Hermitian Eigenvalue Problems

　6.4 Hermitian Matrix Diagonalization

　6.5 NormaIMatrices

　AdditionalReadings

7 Ordinary DifTerential Equations

　7.1 Introduction

　7.2 First-OrderEquations

　7.3 ODEs with Constant Coefficients

　7.4 Second-Order Linear ODEs

　7.5 Series Solutions-Frobenius ' Method

　7.6 OtherSolutions

　7.7 Inhomogeneous Linear ODEs

　7.8 Nonlinear Differential Equations

　Additional Readings

8 Sturm-Liouville Theory

　8.1 Introduction

　8.2 Hermitian Operators

　8.3 ODE Eigenvalue Problems

　8.4 Variation Method

　8.5 Summary, Eigenvalue Problems

　Additional Readings

9 Partial Differential Equations

　9.1 Introduction

　9.2 First-Order Equations

　9.3 Second-Order Equations

　9.4 Separation of Variables

　9.5 Laplace and Poisson Equations

　9.6 Wave Equation

　9.7 Heat-Flow, or Diffusion PDE

　9.8 Summary

　Additional Readings

10 Green's Functions

　10.1 One-Dimensional Problems

　10.2 Problems in Two and Three Dimensions

　Additional Readings

11 Complex Variable Theory

　11.1 Complex Variables and Functions

　11.2 Cauchy-Riemann Conditions

　11.3 Cauchy' s Integral Theorem

　11.4 Cauchy' s Integral Formula

　11.5 Laurent Expansion

　11.6 Singularities

　11.7 Calculus of Residues

　11.8 Evaluation of Definite Integrals

　11.9 Evaluation of Sums

　11.10 Miscellaneous Topics

　Additional Readings

12 Further Topics in Analysis

　12.1 Orthogonal Polynomials

　12.2 Bernoulli Numbers

　12.3 Euler-Maclaurin Integration Formula

　12.4 Dirichlet Series

　12.5 Infinite Products

　12.6 Asymptotic Series

　12.7 Method of Steepest Descents

　12.8 Dispersion Relations

　Additional Readings

13 Gamma Function

　13.1 Definitions, Properties

　13.2 Digamma and Polygamma Functions

　13.3 The Beta Function

　13.4 Stirling's Series

　13.5 Riemann Zeta Function

　13.6 Other Related Functions

　Additional Readings

14 Bessel Functions

　14.1 Bessel Functions of the First Kind, ,Iv (x)

　14.2 Orthogonality

　14.3 Neumann Functions, Bessel Functions of the Second Kind

　14.4 Hankel Functions

　14.5 Modified Bessel Functions, Iv (x) and Kv (x)

　14.6 Asymptotic Expansions

　14.7 Spherical Bessel Functions

　Additional Readings

15 Legendre Functions

　15.1 Legendre Polynomials

　15.2 Orthogonality

　15.3 Physical Interpretation of Generating Function

　15.4 Associated L