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.

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 TheoremChapter 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,f,z) 2.5 Spherical Polar Coordinates (r,0,f) 2.6 Separation of VariablesChapter 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 OperationsChapter 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 GroupChapter 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 ProductsChapter 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 MappingChapter 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 DescentsChapter 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 SolutionsChapter 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 EigenfunctionsChapter 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 FunctionsChapter 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 FunctionsChapter 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 HarmonicsChapter 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 FunctionsChapter 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 TransformChapter 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 TransformationChapter 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 DimensionsChapter 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 TechniqueAppendix 1 Real Zeros of a FunctionAppendix 2 Gaussian QuadratureGeneral ReferencesIndex