MATHEMATICAL PHYSICS

By T. CARIR

CONTENTS ANALYTIC FUNCTIONS COMPLEX NUMBERS MOTIVATION AND DE?NITIONS TRIANGLE INEQUALITIES POLAR REPRESENTATION ARGUMENT FUNCTION TAKE CARE WITH MULTIVALUED FUNCTIONS FUNCTIONS AS MAPPINGS MAPPING: W Z MAPPING: W SINZ ELEMENTARY FUNCTIONS AND THEIR INVERSES EXPONENTIAL AND LOGARITHM POWERS TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS STANDARD BRANCH CUTS SETS, CURVES, REGIONS AND DOMAINS LIMITS AND CONTINUITY DI?ERENTIABILITY CAUCHY–RIEMANN EQUATIONS DI?ERENTIATION RULES PROPERTIES OF ANALYTIC FUNCTIONS CAUCHY–GOURSAT THEOREM SIMPLY CONNECTED REGIONS PROOF EXAMPLE 110 CAUCHY INTEGRAL FORMULA 1101 INTEGRATION AROUND NONANALYTIC REGIONS 1102 CAUCHY INTEGRAL FORMULA 1103 EXAMPLE: YUKAWA FIELD 1104 DERIVATIVES OF ANALYTIC FUNCTIONS 1105 MORERA’S THEOREM 111 COMPLEX SEQUENCES AND SERIES 1111 CONVERGENCE TESTS 1112 UNIFORM CONVERGENCE 112 DERIVATIVES AND TAYLOR SERIES FOR ANALYTIC FUNCTIONS 1121 TAYLOR SERIES 1122 CAUCHY INEQUALITY 1123 LIOUVILLE’S THEOREM 1124 FUNDAMENTAL THEOREM OF ALGEBRA 1125 ZEROS OF ANALYTIC FUNCTIONS 113 LAURENT SERIES 1131 DERIVATION 1132 EXAMPLE 1133 CLASSI?CATION OF SINGULARITIES 1134 POLES AND RESIDUES 114 MEROMORPHIC FUNCTIONS 1141 POLE EXPANSION 1142 EXAMPLE: TANZ 1143 PRODUCT EXPANSION 1144 EXAMPLE: SINZ INTEGRATION INTRODUCTION GOOD TRICKS PARAMETRIC DI?ERENTIATION CONVERGENCE FACTORS CONTOUR INTEGRATION RESIDUE THEOREM DE?NITE INTEGRALS OF THE FORM 2? F SIN ?, COS ? ? DE?NITE INTEGRALS OF THE FORM F X X FOURIER INTEGRALS CUSTOM CONTOURS ISOLATED SINGULARITIES ON THE CONTOUR REMOVABLE SINGULARITY CAUCHY PRINCIPAL VALUE INTEGRATION AROUND A BRANCH POINT REDUCTION TO TABULATED INTEGRALS EXAMPLE: X4 X EXAMPLE: THE BETA FUNCTION EXAMPLE: ?N ?? 1 ? INTEGRAL REPRESENTATIONS FOR ANALYTIC FUNCTIONS USING TO EVALUATE INTEGRALS SYMBOLIC INTEGRATION NUMERICAL INTEGRATION FURTHER INFORMATION ASYMPTOTIC SERIES METHOD OF STEEPEST DESCENT EXAMPLE: GAMMA FUNCTION PARTIAL INTEGRATION EXAMPLE: COMPLEMENTARY ERROR FUNCTION EXPANSION OF AN INTEGRAND EXAMPLE: MODI?ED BESSEL FUNCTION GENERALIZED FUNCTIONS MOTIVATION PROPERTIES OF THE DIRAC DELTA FUNCTION OTHER USEFUL GENERALIZED FUNCTIONS HEAVISIDE STEP FUNCTION DERIVATIVES OF THE DIRAC DELTA FUNCTION GREEN FUNCTIONS MULTIDIMENSIONAL DELTA FUNCTIONS INTEGRAL TRANSFORMS FOURIER TRANSFORM DE?NITION AND INVERSION BASIC PROPERTIES PARSEVAL’S THEOREM CONVOLUTION THEOREM CORRELATION THEOREM USEFUL FOURIER TRANSFORMS FOURIER TRANSFORM OF DERIVATIVES SUMMARY GREEN FUNCTIONS VIA FOURIER TRANSFORM EXAMPLE: GREEN FUNCTION FOR ONE-DIMENSIONAL DI?USION EXAMPLE: THREE-DIMENSIONAL GREEN FUNCTION FOR DI?USION EQUATIONS EXAMPLE: GREEN FUNCTION FOR DAMPED OSCILLATOR OPERATOR METHOD COSINE OR SINE TRANSFORMS FOR EVEN OR ODD FUNCTIONS DISCRETE FOURIER TRANSFORM SAMPLING CONVOLUTION TEMPORAL CORRELATION POWER SPECTRUM ESTIMATION LAPLACE TRANSFORM LAPLACE TRANSFORMS FOR ELEMENTARY FUNCTIONS LAPLACE TRANSFORM OF DERIVATIVES GREEN FUNCTIONS VIA LAPLACE TRANSFORM EXAMPLE: SERIES RC CIRCUIT EXAMPLE: DAMPED OSCILLATOR EXAMPLE: DI?USION WITH CONSTANT BOUNDARY VALUE ANALYTIC CONTINUATION AND DISPERSION RELATIONS ANALYTIC CONTINUATION UNIQUENESS RE?ECTION PRINCIPLE PERMANENCE OF ALGEBRAIC FORM DISPERSION RELATIONS CAUSALITY OSCILLATOR MODEL KRAMERS–KRONIG RELATIONS SUM RULES HILBERT TRANSFORM SPREADING OF A WAVE PACKET SOLITONS STURM–LIOUVILLE THEORY INTRODUCTION: THE GENERAL STRING EQUATION HILBERT SPACES SCHWARTZ INEQUALITY GRAM–SCHMIDT ORTHOGONALIZATION PROPERTIES OF STURM–LIOUVILLE SYSTEMS SELF-ADJOINTNESS REALITY OF EIGENVALUES AND ORTHOGONALITY OF EIGENFUNCTIONS DISCRETENESS OF EIGENVALUES COMPLETENESS OF EIGENFUNCTIONS REALITY OF EIGENFUNCTIONS INTERLEAVING OF ZEROS COMPARISON THEOREMS INTERFACE MATCHING EIGENFUNCTION EXPANSION OF GREEN FUNCTION EXAMPLE: VIBRATING STRING PERTURBATION THEORY EXAMPLE: BEAD AT CENTER OF A STRING VARIATIONAL METHODS LEGENDRE AND BESSEL FUNCTIONS LEGENDRE FUNCTIONS GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS SERIES REPRESENTATION AND RODRIGUES’ FORMULA SCHLÄ?I’S INTEGRAL REPRESENTATION LEGENDRE EXPANSION ASSOCIATED LEGENDRE FUNCTIONS SPHERICAL HARMONICS MULTIPOLE EXPANSION ADDITION THEOREM LEGENDRE FUNCTIONS OF THE SECOND KIND 8210 RELATIONSHIP TO HYPERGEOMETRIC FUNCTIONS 8211 ANALYTIC STRUCTURE OF LEGENDRE FUNCTIONS BESSEL FUNCTIONS CYLINDRICAL HANKEL FUNCTIONS NEUMANN FUNCTIONS MODI?ED BESSEL FUNCTIONS SPHERICAL BESSEL FUNCTIONS FOURIER–BESSEL TRANSFORM EXAMPLE: FOURIER–BESSEL EXPANSION OF NUCLEAR CHARGE DENSITY CYLINDRICAL BESSEL FUNCTIONS FOURIER–BESSEL EXPANSIONS BOUNDARY-VALUE PROBLEMS LAPLACE’S EQUATION IN BOX WITH SPECI?ED POTENTIAL ON ONE SIDE GREEN FUNCTION FOR GROUNDED BOX GREEN’S THEOREM FOR ELECTROSTATICS SEPARABLE COORDINATE SYSTEMS SPHERICAL POLAR COORDINATES CYLINDRICAL COORDINATES SPHERICAL EXPANSION OF DIRICHLET GREEN FUNCTION FOR POISSON’S EQUATION EXAMPLE: MULTIPOLE EXPANSION FOR LOCALIZED CHARGE DISTRIBUTION EXAMPLE: POINT CHARGE NEAR GROUNDED CONDUCTING SPHERE EXAMPLE: SPECI?ED POTENTIAL ON SURFACE OF EMPTY SPHERE EXAMPLE: CHARGED RING AT CENTER OF GROUNDED CONDUCTING SPHERE MAGNETIC FIELD OF CURRENT LOOP INHOMOGENEOUS WAVE EQUATION SPATIAL REPRESENTATION OF TIME-INDEPENDENT GREEN FUNCTION PARTIAL-WAVE EXPANSION MOMENTUM REPRESENTATION OF TIME-INDEPENDENT GREEN FUNCTION RETARDED GREEN FUNCTION LIPPMANN–SCHWINGER EQUATION 10 GROUP THEORY 101 INTRODUCTION 102 FINITE GROUPS 1021 DE?NITIONS 1022 EQUIVALENCE CLASSES 1023 SUBGROUPS 1024 HOMOMORPHISM 1025 DIRECT PRODUCTS 103 REPRESENTATIONS 1031 DE?NITIONS 1032 EXAMPLE: VIBRATING TRIANGLE 1033 ORTHOGONALITY THEOREM 1034 CHARACTER 1035 EXAMPLE: CHARACTER TABLE FOR SYMMETRIES OF A SQUARE 1036 EXAMPLE: VIBRATIONAL EIGENVALUES OF SQUARE 1037 DIRECT-PRODUCT REPRESENTATIONS 1038 EIGENFUNCTIONS 1039 WIGNER–ECKART THEOREM 104 CONTINUOUS GROUPS 1041 DE?NITIONS 1042 TRANSFORMATION OF FUNCTIONS 1043 GENERATORS 1044 EXAMPLE: LINEAR COORDINATE TRANSFORMATIONS IN ONE DIMENSION 1045 EXAMPLE: SO(2) 1046 EXAMPLE: SU(2) 1047 EXAMPLE: SO(3) 1048 TOTAL ANGULAR MOMENTUM 1049 TRANSFORMATION OF OPERATORS 10410 INVARIANT FUNCTIONS 105 LIE ALGEBRA 1051 DE?NITIONS 1052 EXAMPLE: SU(2) 106 ORTHOGONALITY RELATIONS FOR LIE GROUPS 107 QUANTUM MECHANICAL REPRESENTATIONS OF THE ROTATION GROUP 1071 GENERATORS AND COMMUTATION RELATIONS 1072 EULER PARAMETRIZATION 1073 HOMOMORPHISM BETWEEN SU(2) AND SO(3) 1074 IRREDUCIBLE REPRESENTATIONS OF SU(2) 1075 ORTHOGONALITY RELATIONS FOR ROTATION MATRICES 1076 COUPLING OF ANGULAR MOMENTA 1077 SPHERICAL TENSORS 108 UNITARY SYMMETRIES IN NUCLEAR AND PARTICLE PHYSICS