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Michael T. VaughnIntroduction to Mathematical AL1.1.Z;rWILEY-VCH Verlag GmbH & Co. KGaA

Contents1 Infinite Sequences and Series1.1 Real and Complex Numbers1.1.1 Arithmetic1.1.2 Algebraic Equations1.1.3 Infinite Sequences; Irrational Numbers1.1.4 Sets of Real and Complex Numbers1.2 Convergence of Infinite Series and Products1.2.1 Convergence and Divergence; Absolute Convergence1.2.2 Tests for Convergence of an Infinite Series of Positive Terms1.2.3 Alternating Series and Rearrangements1.2.4 Infinite Products1.3 Sequences and Series of Functions1.3.1 Pointwise Convergence and Uniform Convergence of Sequences ofFunctions1.3.2 Weak Convergence; Generalized Functions1.3.3 Infinite Series of Functions; Power Series1.4 Asymptotic Series1.4.1 The Exponential Integral1.4.2 Asymptotic Expansions; Asymptotic Series1.4.3 Laplace Integral; Watson's LemmaA Iterated Maps, Period Doubling, and ChaosBibliography and NotesProblems141516191920222630312 Finite-Dimensional Vector Spaces2.1 Linear Vector Spaces2.1.1 Linear Vector Space Axioms2.1.2 Vector Norm; Scalar Product2.1.3 Sum and Product Spaces2.1.4 Sequences of Vectors2.1.5 Linear Functionals and Dual Spaces2.2 Linear Operators2.2.1 Linear Operators; Domain and Image; Bounded Operators2.2.2 Matrix Representation; Multiplication of Linear Operators37414143474949515154huroduction to Mathematicai Physics. Michael T. VaughnCopyright 2007 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-40627-21334578810111314

ContentsVI2.2.3 The Adjoint Operator2.2.4 Change of Basis; Rotations; Unitary Operators2.2.5 Invariant Manifolds2.2.6 Projection Operators2.3 Eigenvectors and Eigenvalues2.3.1 Eigenvalue Equation2.3.2 Diagonalization of a Linear Operator2.3.3 Spectral Representation of Normal Operators2.3.4 Minimax Properties of Eigenvalues of Self-Adjoint Operators2.4 Functions of Operators2.5 Linear Dynamical SystemsSmall OscillationsABibliography and NotesProblems3 Geometry in Physics3.1 Manifolds and Coordinates3.1.1 Coordinates on Manifolds3.1.2 Some Elementary Manifolds3.1.3 Elementary Properties of Manifolds3.2 Vectors, Differential Forms, and Tensors3.2.1 Smooth Curves and Tangent Vectors3.2.2 Tangent Spaces and the Tangent Bundle T(.A4)3.2.3 Differential Forms3.2.4 Tensors3.2.5 Vector and Tensor Fields3.2.6 The Lie Derivative3.3 Calculus on Manifolds3.3.1 Wedge Product: p-Forms and p-Vectors3.3.2 Exterior Derivative3.3.3 Stokes' Theorem and its Generalizations3.3.4 Closed and Exact Forms3.4 Metric Tensor and Distance3.4.1 Metric Tensor of a Linear Vector Space3.4.2 Raising and Lowering Indices3.4.3 Metric Tensor of a Manifold3.4.4 Metric Tensor and Volume3.4.5 The Laplacian Operator3.4.6 Geodesic Curves on a Manifold3.5 Dynamical Systems and Vector Fields3.5.1 What is a Dynamical System?3.5.2 A Model from Ecology3.5.3 Lagrangian and Hamiltonian Systems3.6 Fluid 34135139139140142148

ContentsACalculus of VariationsB ThermodynamicsBibliography and NotesProblemsVII1521531581594 Functions of a Complex Variable4.1 Elementary Properties of Analytic Functions4.1.1 Cauchy—Riemann Conditions4.1.2 Conformal Mappings4.2 Integration in the Complex Plane4.2.1 Integration Along a Contour4.2.2 Cauchy's Theorem4.2.3 Cauchy's Integral Formula4.3 Analytic Functions4.3.1 Analytic Continuation4.3.2 Singularities of an Analytic Function4.3.3 Global Properties of Analytic Functions4.3.4 Laurent Series4.3.5 Infinite Product Representations4.4 Calculus of Residues: Applications4.4.1 Cauchy Residue Theorem4.4.2 Evaluation of Real Integrals4.5 Periodic Functions; Fourier Series4.5.1 Periodic Functions4.5.2 Doubly Periodic FunctionsA Gamma Function; Beta FunctionGamma FunctionA.1Beta FunctionA.2Bibliography and 861881901901911951951971991992032042055 Differential Equations: Analytical Methods5.1 Systems of Differential Equations5.1.1 General Systems of First-Order Equations5.1.2 Special Systems of Equations5.2 First-Order Differential Equations5.2.1 Linear First-Order Equations5.2.2 Ricatti Equation5.2.3 Exact Differentials5.3 Linear Differential Equations5.3.1 nth Order Linear Equations5.3.2 Power Series Solutions5.3.3 Linear Independence; General Solution5.3.4 Linear Equation with Constant 5

ContentsVIII5.4 Linear Second-Order Equations5.4.1 Classification of Singular Points5.4.2 Exponents at a Regular Singular Point5.4.3 One Regular Singular Point5.4.4 Two Regular Singular Points5.5 Legendre's Equation5.5.1 Legendre Polynomials5.5.2 Legendre Functions of the Second Kind5.6 Bessel's Equation5.6.1 Bessel Functions5.6.2 Hankel Functions5.6.3 Spherical Bessel FunctionsA Hypergeometric EquationReduction to Standard FormA.1Power Series SolutionsA.2Integral RepresentationsA.3B Confluent Hypergeometric EquationB.1Reduction to Standard FormB.2Integral RepresentationsCElliptic Integrals and Elliptic FunctionsBibliography and NotesProblems6 Hilbert Spaces6.1 Infinite-Dimensional Vector Spaces6.1.1 Hilbert Space Axioms6.1.2 Convergence in Hilbert space6.2 Function Spaces; Measure Theory6.2.1 Polynomial Approximation; Weierstrass Approximation Theorem6.2.2 Convergence in the Mean6.2.3 Measure Theory6.3 Fourier Series6.3.1 Periodic Functions and Trigonometrie Polynomials6.3.2 Classical Fourier Series6.3.3 Convergence of Fourier Series6.3.4 Fourier Cosine Series; Fourier Sine Series6.4 Fourier Integral; Integral Transforms6.4.1 Fourier Transform6.4.2 Convolution Theorem; Correlation Functions6.4.3 Laplace Transform6.4.4 Multidimensional Fourier Transform6.4.5 Fourier Transform in Quantum Mechanics6.5 Orthogonal Polynomials6.5.1 Weight Functions and Orthogonal Polynomials6.5.2 Legendre Polynomials and Associated Legendre Functions6.5.3 Spherical 73273274275279281281284286287288289289290292

Contents6.6 Haar Functions; WaveletsA Standard Families of Orthogonal PolynomialsBibliography and NotesProblemsIX2943053103117 Linear Operators on Hilbert Space7.1 Some Hilbert Space Subtleties7.2 General Properties of Linear Operators on Hilbert Space7.2.1 Bounded, Continuous, and Closed Operators7.2.2 Inverse Operator7.2.3 Compact Operators; Hilbert–Schmidt Operators7.2.4 Adjoint Operator7.2.5 Unitary Operators; Isometric Operators7.2.6 Convergence of Sequences of Operators in 7-17.3 Spectrum of Linear Operators on Hilbert Space7.3.1 Spectrum of a Compact Self-Adjoint Operator7.3.2 Spectrum of Noncompact Normal Operators7.3.3 Resolution of the Identity7.3.4 Functions of a Self-Adjoint Operator7.4 Linear Differential Operators7.4.1 Differential Operators and Boundary Conditions7.4.2 Second-Order Linear Differential Operators7.5 Linear Integral Operators; Green Functions7.5.1 Compact Integral Operators7.5.2 Differential Operators and Green FunctionsBibliography and 323353363363383393393413443458 Partial Differential Equations8.1 Linear First-Order Equations8.2 The Laplacian and Linear Second-Order Equations8.2.1 Laplacian and Boundary Conditions8.2.2 Green Functions for Laplace's Equation8.2.3 Spectrum of the Laplacian8.3 Time-Dependent Partial Differential Equations8.3.1 The Diffusion Equation8.3.2 Inhomogeneous Wave Equation: Advanced and Retarded GreenFunctions8.3.3 The Schrödinger Equation8.4 Nonlinear Partial Differential Equations8.4.1 Quasilinear First-Order Equations8.4.2 KdV Equation8.4.3 Scalar Field in 1 1 Dimensions8.4.4 Sine-Gordon EquationA Lagrangian Field 3384

ContentsXBibliography and NotesProblems9 Finite Groups9.1 General Properties of Groups9.1.1 Group Axioms9.1.2 Cosets and Classes9.1.3 Algebras; Group Algebra9.2 Some Finite Groups9.2.1 Cyclic Groups9.2.2 Dihedral Groups9.2.3 Tetrahedral Group9.3 The Symmetrie Group SN9.3.1 Permutations and the Symmetrie Group SN9.3.2 Permutations and Partitions9.4 Group Representations9.4.1 Group Representations by Linear Operators9.4.2 Schur's Lemmas and Orthogonality Relations9.4.3 Kronecker Product of Representations9.4.4 Permutation Representations9.4.5 Representations of Groups and Subgroups9.5 Representations of the Symmetrie Group SN9.5.1 Irreducible Representations of SN9.5.2 Outer Products of Representations of Sm ST,9.5.3 Kronecker Products of Irreducible Representations of SN9.6 Discrete Infinite GroupsA Frobenius Reciprocity TheoremBS-Functions and Irreducible Representations of SNB.1Frobenius Generating Funetion for the Simple Characters of SNB.2Graphical Calculation of the Characters x ))(,B.3Outer Products of Representations of Sm (97,Bibliography and NotesProblems10 Lie Groups and Lie Algebras10.1 Lie Groups10.2 Lie Algebras10.2.1 The Generators of a Lie Group10.2.2 The Lie Algebra of a Lie Group10.2.3 Classification of Lie Algebras10.3 Representations of Lie Algebras10.3.1 Irreducible Representations of SU(2)10.3.2 Addition of Angular Momenta10.3.3 SN and the Irreducible Representations of SU(2)10.3.4 Irreducible Representations of 1457460461461462465469469471474476

ContentsA Tensor Representations of the Classical Lie GroupsA.1The Classical Lie GroupsA.2Tensor Representations of U(n) and SU (n)A.3Irreducible Representations of SO (n)B Lorentz Group; Poincar6 GroupB.1Lorentz TransformationsB.2S L(2, C) and the Homogeneous Lorentz GroupB.3Inhomogeneous Lorentz Transformations; Poincar6 GroupBibliography and 507

3 Geometry in Physics 93 3.1 Manifolds and Coordinates 97 3.1.1 Coordinates on Manifolds 97 3.1.2 Some Elementary Manifolds 98 3.1.3 Elementary Properties of Manifolds 101 3.2 Vectors, Differential Forms, and Tensors 104 3.2.1 Smooth Curves and Tangent Vectors 104 3.2.2 Tangent Spaces and the Tangent Bundle T(.A4) 105