# Mathematical Methods for Physicists

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Mathematical Methods for Physicists, Third Edition provides an advanced undergraduate and beginning graduate study in physical science, focusing on the mathematics of theoretical physics. This edition includes sections on the non-Cartesian tensors, dispersion theory, first-order differential equations, numerical application of Chebyshev polynomials, the fast Fourier transform, and transfer functions. Many of the physical examples provided in this book, which are used to illustrate the applications of mathematics, are taken from the fields of electromagnetic theory and quantum mechanics. The Hermitian operators, Hilbert space, and concept of completeness are also deliberated. This book is beneficial to students studying graduate level physics, particularly theoretical physics.

## Table of Contents

Chapter 1 Vector Analysis

1.1 Definitions, Elementary Approach

1.2 Advanced Definitions

1.3 Scalar or Dot Product

1.4 Vector or Cross Product

1.5 Triple Scalar Product, Triple Vector Product

1.6 Gradient

1.7 Divergence

1.8 Curl

1.9 Successive Applications of V

1.10 Vector Integration

1.11 Gauss's Theorem

1.12 Stokes's Theorem

1.13 Potential Theory

1.14 Gauss's Law, Poisson's Equation

1.15 Helmholtz's Theorem

Chapter 2 Coordinate Systems

2.1 Curvilinear Coordinates

2.2 Differential Vector Operations

2.3 Special Coordinate Systems—Rectangular Cartesian Coordinates

2.4 Circular Cylindrical Coordinates (p,φ,z)

2.5 Spherical Polar Coordinates (r,0,φ)

2.6 Separation of Variables

Chapter 3 Tensor Analysis

3.1 Introduction, Definitions

3.2 Contraction, Direct Product

3.3 Quotient Rule

3.4 Pseudotensors, Dual Tensors

3.5 Dyadics

3.6 Theory of Elasticity

3.7 Lorentz Co variance of Maxwell's Equations

3.8 Noncartesian Tensors, Co variant Differentiation

3.9 Tensor Differential Operations

Chapter 4 Determinants, Matrices, and Group Theory

4.1 Determinants

4.2 Matrices

4.3 Orthogonal Matrices

4.4 Oblique Coordinates

4.5 Hermitian Matrices, Unitary Matrices

4.6 Diagonalization of Matrices

4.7 Eigenvectors, Eigenvalues

4.8 Introduction to Group Theory

4.9 Discrete Groups

4.10 Continuous Groups

4.11 Generators

4.12 SU(2), SU(3), and Nuclear Particles

4.13 Homogeneous Lorentz Group

Chapter 5 Infinite Series

5.1 Fundamental Concepts

5.2 Convergence Tests

5.3 Alternating Series

5.4 Algebra of Series

5.5 Series of Functions

5.6 Taylor's Expansion

5.7 Power Series

5.8 Elliptic Integrals

5.9 Bernoulli Numbers, Euler-Maclaurin Formula

5.10 Asymptotic or Semiconvergent Series

5.11 Infinite Products

Chapter 6 Functions of a Complex Variable I

6.1 Complex Algebra

6.2 Cauchy-Riemann Conditions

6.3 Cauchy's Integral Theorem

6.4 Cauchy's Integral Formula

6.5 Laurent Expansion

6.6 Mapping

6.7 Conformal Mapping

Chapter 7 Functions of a Complex Variable II: Calculus of Residues 396

7.1 Singularities

7.2 Calculus of Residues

7.3 Dispersion Relations

7.4 The Method of Steepest Descents

Chapter 8 Differential Equations

8.1 Partial Differential Equations of Theoretical Physics

8.2 First-Order Differential Equations

8.3 Separation of Variables—Ordinary Differential Equations

8.4 Singular Points

8.5 Series Solutions—Frobenius Method

8.6 A Second Solution

8.7 Nonhomogeneous Equation—Green's Function

8.8 Numerical Solutions

Chapter 9 Sturm-Liouville Theory - Orthogonal Functions

9.1 Self-Adjoint Differential Equations

9.2 Hermitian (Self-Adjoint) Operators

9.3 Gram-Schmidt Orthogonalization

9.4 Completeness of Eigenfunctions

Chapter 10 The Gamma Function (Factorial Function)

10.1 Definitions, Simple Properties

10.2 Digamma and Polygamma Functions

10.3 Stirling's Series

10.4 The Beta Function

10.5 The Incomplete Gamma Functions and Related Functions

Chapter 11 Bessel Functions

11.1 Bessel Functions of the First Kind, Jv(x)

11.2 Orthogonality

11.3 Neumann Functions, Bessel Functions of the Second Kind, Nv(x)

11.4 Hankel Functions

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

11.6 Asymptotic Expansions

11.7 Spherical Bessel Functions

Chapter 12 Legendre Functions

12.1 Generating Function

12.2 Recurrence Relations and Special Properties

12.3 Orthogonality

12.4 Alternate Definitions of Legendre Polynomials

12.5 Associated Legendre Functions

12.6 Spherical Harmonics

12.7 Angular Momentum Ladder Operators

12.8 The Addition Theorem for Spherical Harmonics

12.9 Integrals of the Product of Three Spherical Harmonics

12.10 Legendre Functions of the Second Kind, Qn(x)

12.11 Vector Spherical Harmonics

Chapter 13 Special Functions

13.1 Hermite Functions

13.2 Laguerre Functions

13.3 Chebyshev (Tschebyscheff) Polynomials

13.4 Chebyshev Polynomials—Numerical Applications

13.5 Hypergeometric Functions

13.6 Confluent Hypergeometric Functions

Chapter 14 Fourier Series

14.1 General Properties

14.2 Advantages, Uses of Fourier Series

14.3 Applications of Fourier Series

14.4 Properties of Fourier Series

14.5 Gibbs Phenomenon

14.6 Discrete Orthogonality—Discrete Fourier Transform

Chapter 15 Integral Transforms

15.1 Integral Transforms

15.2 Development of the Fourier Integral

15.3 Fourier Transforms—Inversion Theorem

15.4 Fourier Transform of Derivatives

15.5 Convolution Theorem

15.6 Momentum Representation

15.7 Transfer Functions

15.8 Elementary Laplace Transforms

15.9 Laplace Transform of Derivatives

15.10 Other Properties

15.11 Convolution or Faltung Theorem

15.12 Inverse Laplace Transformation

Chapter 16 Integral Equations

16.1 Introduction

16.2 Integral Transforms, Generating Functions

16.3 Neumann Series, Separable (Degenerate) Kernels

16.4 Hilbert-Schmidt Theory

16.5 Green's Functions—One Dimension

16.6 Green's Functions—Two and Three Dimensions

Chapter 17 Calculus of Variations

17.1 One-Dependent and One-Independent Variable

17.2 Applications of the Euler Equation

17.3 Generalizations, Several Dependent Variables

17.4 Several Independent Variables

17.5 More Than One Dependent, More than One Independent Variable

17.6 Lagrangian Multipliers

17.7 Variation Subject to Constraints

17.8 Rayleigh-Ritz Variational Technique

Appendix 1 Real Zeros of a Function

Appendix 2 Gaussian Quadrature

General References

Index

## Product details

- No. of pages: 1008
- Language: English
- Copyright: © Academic Press 1985
- Published: January 1, 1985
- Imprint: Academic Press
- eBook ISBN: 9781483277820

## About the Author

### George Arfken

#### Affiliations and Expertise

Miami University, Oxford, Ohio, USA

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