AN INTRODUCTION TO MATHEMATICAL PHYSICS
CONTENTS IN?NITE SEQUENCES AND SERIES REAL AND COMPLEX NUMBERS ARITHMETIC ALGEBRAIC EQUATIONS IN?NITE SEQUENCES; IRRATIONAL NUMBERS SETS OF REAL AND COMPLEX NUMBERS CONVERGENCE OF IN?NITE SERIES AND PRODUCTS CONVERGENCE AND DIVERGENCE; ABSOLUTE CONVERGENCE TESTS FOR CONVERGENCE OF AN IN?NITE SERIES OF POSITIVE TERMS ALTERNATING SERIES AND REARRANGEMENTS IN?NITE PRODUCTS SEQUENCES AND SERIES OF FUNCTIONS POINTWISE CONVERGENCE AND UNIFORM CONVERGENCE OF SEQUENCES OF FUNCTIONS WEAK CONVERGENCE; GENERALIZED FUNCTIONS IN?NITE SERIES OF FUNCTIONS; POWER SERIES ASYMPTOTIC SERIES THE EXPONENTIAL INTEGRAL ASYMPTOTIC EXPANSIONS; ASYMPTOTIC SERIES LAPLACE INTEGRAL; WATSON’S LEMMA A ITERATED MAPS, PERIOD DOUBLING, AND CHAOS BIBLIOGRAPHY AND NOTES PROBLEMS FINITE-DIMENSIONAL VECTOR SPACES LINEAR VECTOR SPACES LINEAR VECTOR SPACE AXIOMS VECTOR NORM; SCALAR PRODUCT SUM AND PRODUCT SPACES SEQUENCES OF VECTORS LINEAR FUNCTIONALS AND DUAL SPACES LINEAR OPERATORS LINEAR OPERATORS; DOMAIN AND IMAGE; BOUNDED OPERATORS MATRIX REPRESENTATION; MULTIPLICATION OF LINEAR OPERATORS THE ADJOINT OPERATOR CHANGE OF BASIS; ROTATIONS; UNITARY OPERATORS INVARIANT MANIFOLDS PROJECTION OPERATORS EIGENVECTORS AND EIGENVALUES EIGENVALUE EQUATION DIAGONALIZATION OF A LINEAR OPERATOR SPECTRAL REPRESENTATION OF NORMAL OPERATORS MINIMAX PROPERTIES OF EIGENVALUES OF SELF-ADJOINT OPERATORS FUNCTIONS OF OPERATORS LINEAR DYNAMICAL SYSTEMS A SMALL OSCILLATIONS GEOMETRY IN PHYSICS MANIFOLDS AND COORDINATES COORDINATES ON MANIFOLDS SOME ELEMENTARY MANIFOLDS ELEMENTARY PROPERTIES OF MANIFOLDS VECTORS, DIFFERENTIAL FORMS, AND TENSORS SMOOTH CURVES AND TANGENT VECTORS TANGENT SPACES AND THE TANGENT BUNDLE T (M) DIFFERENTIAL FORMS TENSORS VECTOR AND TENSOR FIELDS THE LIE DERIVATIVE CALCULUS ON MANIFOLDS WEDGE PRODUCT: P-FORMS AND P-VECTORS EXTERIOR DERIVATIVE STOKES’ THEOREM AND ITS GENERALIZATIONS CLOSED AND EXACT FORMS METRIC TENSOR AND DISTANCE METRIC TENSOR OF A LINEAR VECTOR SPACE RAISING AND LOWERING INDICES METRIC TENSOR OF A MANIFOLD METRIC TENSOR AND VOLUME THE LAPLACIAN OPERATOR GEODESIC CURVES ON A MANIFOLD DYNAMICAL SYSTEMS AND VECTOR FIELDS WHAT IS A DYNAMICAL SYSTEM? A MODEL FROM ECOLOGY LAGRANGIAN AND HAMILTONIAN SYSTEMS FLUID MECHANICS A CALCULUS OF VARIATIONS B THERMODYNAMICS FUNCTIONS OF A COMPLEX VARIABLE ELEMENTARY PROPERTIES OF ANALYTIC FUNCTIONS CAUCHY–RIEMANN CONDITIONS CONFORMAL MAPPINGS INTEGRATION IN THE COMPLEX PLANE INTEGRATION ALONG A CONTOUR CAUCHY’S THEOREM CAUCHY’S INTEGRAL FORMULA ANALYTIC FUNCTIONS ANALYTIC CONTINUATION SINGULARITIES OF AN ANALYTIC FUNCTION GLOBAL PROPERTIES OF ANALYTIC FUNCTIONS LAURENT SERIES IN?NITE PRODUCT REPRESENTATIONS CALCULUS OF RESIDUES: APPLICATIONS CAUCHY RESIDUE THEOREM EVALUATION OF REAL INTEGRALS PERIODIC FUNCTIONS; FOURIER SERIES PERIODIC FUNCTIONS DOUBLY PERIODIC FUNCTIONS A GAMMA FUNCTION; BETA FUNCTION GAMMA FUNCTION BETA FUNCTION DIFFERENTIAL EQUATIONS: ANALYTICAL METHODS SYSTEMS OF DIFFERENTIAL EQUATIONS GENERAL SYSTEMS OF FIRST-ORDER EQUATIONS SPECIAL SYSTEMS OF EQUATIONS FIRST-ORDER DIFFERENTIAL EQUATIONS LINEAR FIRST-ORDER EQUATIONS RICATTI EQUATION EXACT DIFFERENTIALS LINEAR DIFFERENTIAL EQUATIONS NTH ORDER LINEAR EQUATIONS POWER SERIES SOLUTIONS LINEAR INDEPENDENCE; GENERAL SOLUTION LINEAR EQUATION WITH CONSTANT COEF?CIENTS LINEAR SECOND-ORDER EQUATIONS CLASSI?CATION OF SINGULAR POINTS EXPONENTS AT A REGULAR SINGULAR POINT ONE REGULAR SINGULAR POINT TWO REGULAR SINGULAR POINTS LEGENDRE’S EQUATION LEGENDRE POLYNOMIALS LEGENDRE FUNCTIONS OF THE SECOND KIND BESSEL’S EQUATION BESSEL FUNCTIONS HANKEL FUNCTIONS SPHERICAL BESSEL FUNCTIONS A HYPERGEOMETRIC EQUATION REDUCTION TO STANDARD FORM INTEGRAL REPRESENTATIONS B CON?UENT HYPERGEOMETRIC EQUATION C ELLIPTIC INTEGRALS AND ELLIPTIC FUNCTIONS HILBERT SPACES IN?NITE-DIMENSIONAL VECTOR SPACES HILBERT SPACE AXIOMS CONVERGENCE IN HILBERT SPACE FUNCTION SPACES; MEASURE THEORY POLYNOMIAL APPROXIMATION; WEIERSTRASS APPROXIMATION THEOREM CONVERGENCE IN THE MEAN MEASURE THEORY FOURIER SERIES PERIODIC FUNCTIONS AND TRIGONOMETRIC POLYNOMIALS CLASSICAL FOURIER SERIES CONVERGENCE OF FOURIER SERIES FOURIER COSINE SERIES; FOURIER SINE SERIES FOURIER INTEGRAL; INTEGRAL TRANSFORMS FOURIER TRANSFORM CONVOLUTION THEOREM; CORRELATION FUNCTIONS LAPLACE TRANSFORM MULTIDIMENSIONAL FOURIER TRANSFORM FOURIER TRANSFORM IN QUANTUM MECHANICS ORTHOGONAL POLYNOMIALS WEIGHT FUNCTIONS AND ORTHOGONAL POLYNOMIALS LEGENDRE POLYNOMIALS AND ASSOCIATED LEGENDRE FUNCTIONS SPHERICAL HARMONICS HAAR FUNCTIONS; WAVELETS A STANDARD FAMILIES OF ORTHOGONAL POLYNOMIALS LINEAR OPERATORS ON HILBERT SPACE SOME HILBERT SPACE SUBTLETIES GENERAL PROPERTIES OF LINEAR OPERATORS ON HILBERT SPACE BOUNDED, CONTINUOUS, AND CLOSED OPERATORS INVERSE OPERATOR COMPACT OPERATORS; HILBERT–SCHMIDT OPERATORS ADJOINT OPERATOR UNITARY OPERATORS; ISOMETRIC OPERATORS CONVERGENCE OF SEQUENCES OF OPERATORS IN H SPECTRUM OF LINEAR OPERATORS ON HILBERT SPACE SPECTRUM OF A COMPACT SELF-ADJOINT OPERATOR SPECTRUM OF NONCOMPACT NORMAL OPERATORS RESOLUTION OF THE IDENTITY FUNCTIONS OF A SELF-ADJOINT OPERATOR LINEAR DIFFERENTIAL OPERATORS DIFFERENTIAL OPERATORS AND BOUNDARY CONDITIONS SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS LINEAR INTEGRAL OPERATORS; GREEN FUNCTIONS COMPACT INTEGRAL OPERATORS DIFFERENTIAL OPERATORS AND GREEN FUNCTIONS PARTIAL DIFFERENTIAL EQUATIONS THE LAPLACIAN AND LINEAR SECOND-ORDER EQUATIONS LAPLACIAN AND BOUNDARY CONDITIONS GREEN FUNCTIONS FOR LAPLACE’S EQUATION SPECTRUM OF THE LAPLACIAN TIME-DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS THE DIFFUSION EQUATION INHOMOGENEOUS WAVE EQUATION: ADVANCED AND RETARDED GREEN FUNCTIONS THE SCHRÖDINGER EQUATION NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS QUASILINEAR FIRST-ORDER EQUATIONS KDV EQUATION SCALAR FIELD IN 1 + 1 DIMENSIONS SINE-GORDON EQUATION A LAGRANGIAN FIELD THEORY FINITE GROUPS GENERAL PROPERTIES OF GROUPS GROUP AXIOMS COSETS AND CLASSES ALGEBRAS; GROUP ALGEBRA SOME FINITE GROUPS CYCLIC GROUPS DIHEDRAL GROUPS TETRAHEDRAL GROUP THE SYMMETRIC GROUP SN PERMUTATIONS AND THE SYMMETRIC GROUP SN PERMUTATIONS AND PARTITIONS GROUP REPRESENTATIONS GROUP REPRESENTATIONS BY LINEAR OPERATORS SCHUR’S LEMMAS AND ORTHOGONALITY RELATIONS KRONECKER PRODUCT OF REPRESENTATIONS PERMUTATION REPRESENTATIONS REPRESENTATIONS OF GROUPS AND SUBGROUPS REPRESENTATIONS OF THE SYMMETRIC GROUP SN IRREDUCIBLE REPRESENTATIONS OF SN OUTER PRODUCTS OF REPRESENTATIONS OF SM ?SN KRONECKER PRODUCTS OF IRREDUCIBLE REPRESENTATIONS OF SN DISCRETE IN?NITE GROUPS A FROBENIUS RECIPROCITY THEOREM B S-FUNCTIONS AND IRREDUCIBLE REPRESENTATIONS OF SN FROBENIUS GENERATING FUNCTION FOR THE SIMPLE CHARACTERS OF SN GRAPHICAL CALCULATION OF THE CHARACTERS ? (?) (M) 10 LIE GROUPS AND LIE ALGEBRAS 10 1 LIE GROUPS 10 2 LIE ALGEBRAS 10 2 1 THE GENERATORS OF A LIE GROUP 10 2 2 THE LIE ALGEBRA OF A LIE GROUP 10 2 3 CLASSI?CATION OF LIE ALGEBRAS 10 3 REPRESENTATIONS OF LIE ALGEBRAS 10 3 1 IRREDUCIBLE REPRESENTATIONS OF SU(2) 10 3 2 ADDITION OF ANGULAR MOMENTA 10 3 3 SN AND THE IRREDUCIBLE REPRESENTATIONS OF SU(2) 10 3 4 IRREDUCIBLE REPRESENTATIONS OF SU(3)