PREFACE
CHAPTER 1. VECTOR ANALYSIS
1.1 DEFINITIONS, ELEMENTARY APPROACH
1.2 ROTATION OF THE COORDINATE AXES3
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 ¿
1.10 VECTOR INTEGRATION
1.11 Gauss’ THEOREM
1.12 STOKES’ THEOREM
1.13 POTENTIAL THEORY
1.14 Gauss’ LAW, POISSON’s EQUATION
1.15 DIRAC DELTA FUNCTION
1.16 HELMHOLTZ’S THEOREM
Additional Readings
CHAPTER 2. VECTOR ANALYSIS IN CURVED COORDINATES AND TENSORS
2.1 ORTHOGONAL COORDINATES IN 3
2.2 DIFFERENTIAL VECTOR OPERATORS
2.3 SPECIAL COORDINATE SYSTEMS: INTRODUCTION
2.4 CIRCULAR CYLINDER COORDINATES
2.5 SPHERICAL POLAR COORDINATES
2.6 TENSOR ANALYSIS
2.7 CONTRACTION, DIRECT PRODUCT
2.8 QUOTIENT RULE
2.9 PSEUDOTENSORS, DUAL TENSORS
2.10 GENERAL TENSORS
2.11 TENSOR DERIVATIVE OPERATORS
Additional Readings
CHAPTER 3. DETERMINANTS AND MATRICES
3.1 DETERMINANTS
3.2 MATRICES
3.3 ORTHOGONAL MATRICES
3.4 HERMITIAN MATRICES, UNITARY MATRICES
3.5 DIAGONALIZATION OF MATRICES
3.6 NORMAL MATRICES
Additional Readings
CHAPTER 4. GROUP THEORY
4.1 INTRODUCTION TO GROUP THEORY
4.2 GENERATORS OF CONTINUOUS GROUPS
4.3 ORBITAL ANGULAR MOMENTUM
4.4 ANGULAR MOMENTUM COUPLING
4.5 HOMOGENEOUS LORENTZ GROUP
4.6 LORENTZ COVARIANCE OF MAXWELL’S EQUATIONS
4.7 DISCRETE GROUPS
4.8 DIFFERENTIAL FORMS
Additional Readings
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 SERIES
5.11 INFINITE PRODUCTS
Additional Readings
CHAPTER 6. FUNCTIONS OF A COMPLEX VARIABLE I ANALYTIC PROPERTIES, MAPPING
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 SINGULARITIES
6.7 MAPPING
6.8 CONFORMAL MAPPING
Additional Readings
CHAPTER 7. FUNCTIONS OF A COMPLEX VARIABLE II
7.1 CALCULUS OF RESIDUES
7.2 DISPERSION RELATIONS
7.3 METHOD OF STEEPEST DESCENTS
Additional Readings
CHAPTER 8. THE GAMMA FUNCTION (FACTORIAL FUNCTION)
8.1 DEFINITIONS, SIMPLE PROPERTIES
8.2 DIGAMMA AND POLYGAMMA FUNCTIONS
8.3 STIRLING’S SERIES
8.4 THE BETA FUNCTION
8.5 THE INCOMPLETE GAMMA FUNCTIONS AND RELATED FUNCTIONS
Additional Readings
CHAPTER 9. DIFFERENTIAL EQUATIONS
9.1 Partial Differential Equations
9.2 First-Order Differential Equations
9.3. SEPARATION OF VARIABLES
9.4 Singular Points
9.5 Series Solutions—Frobenius’ Method
9.8 Heat Flow, or Diffusion, PDE
Additional Readings
CHAPTER 10. STURM-LIOUVILLE THEORY—ORTHOGONAL FUNCTIONS
10.1 SELF-ADJOINT ODES
10.2 HERMITIAN OPERATORS
10.3 GRAM–SCHMIDT ORTHOGONALIZATION
10.4 COMPLETENESS OF EIGENFUNCTIONS
10.5 GREEN’S FUNCTION—EIGENFUNCTION EXPANSION
Additional Readings
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
11.4 HANKEL FUNCTIONS
11.5 MODIFIED BESSEL FUNCTIONS, Iv(x) AND Kv(x)
11.6 ASYMPTOTIC EXPANSIONS
11.7 SPHERICAL BESSEL FUNCTIONS
Additional Readings
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 ORBITAL ANGULAR MOMENTUM OPERATORS
12.8 THE ADDITION THEOREM FOR SPHERICAL HARMONICS
12.9 INTEGRALS OF PRODUCTS OF THREE SPHERICAL HARMONICS
12.10 LEGENDRE FUNCTIONS OF THE SECOND KIND
12.11 VECTOR SPHERICAL HARMONICS
Additional Readings
CHAPTER 13. MORE SPECIAL FUNCTIONS
13.1 HERMITE FUNCTIONS
13.2 LAGUERRE FUNCTIONS
13.3 CHEBYSHEV POLYNOMIALS
13.4 HYPERGEOMETRIC FUNCTIONS
13.5 CONFLUENT HYPERGEOMETRIC FUNCTIONS
13.6 MATHIEU FUNCTIONS
Additional Readings
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 FOURIER TRANSFORM
14.7 FOURIER EXPANSIONS OF MATHIEU FUNCTIONS
Additional Readings
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. LAPLACE TRANSFORMS
15.9. LAPLACE TRANSFORM OF DERIVATIVES
15.10. OTHER PROPERTIES
15.11. CONVOLUTION (FALTUNGS) THEOREM
15.12. INVERSE LAPLACE TRANSFORM
Additional Readings
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
Additional Readings
CHAPTER 17. CALCULUS OF VARIATIONS
17.1 A DEPENDENT AND AN INDEPENDENT VARIABLE
17.2 APPLICATIONS OF THE EULER EQUATION
17.3 SEVERAL DEPENDENT VARIABLES
17.4 SEVERAL INDEPENDENT VARIABLES
17.5 SEVERAL DEPENDENT AND INDEPENDENT VARIABLES
17.6 LAGRANGIAN MULTIPLIERS
17.7 VARIATION WITH CONSTRAINTS
17.8 RAYLEIGH–RITZ VARIATIONAL TECHNIQUE
Additional Readings
CHAPTER 18. NONLINEAR METHODS AND CHAOS
18.1 INTRODUCTION
18.2 THE LOGISTIC MAP
18.3 SENSITIVITY TO INITIAL CONDITIONS AND PARAMETERS
18.4 NONLINEAR DIFFERENTIAL EQUATIONS
Additional Readings
CHAPTER 19. PROBABILITY
19.1 DEFINITIONS, SIMPLE PROPERTIES
19.2 RANDOM VARIABLES
19.3 BINOMIAL DISTRIBUTION
19.4 POISSON DISTRIBUTION
19.5 GAUSS’ NORMAL DISTRIBUTION
19.6 STATISTICS
Additional Readings
INDEX
