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ISBN 10:0521679710
ISBN 13:9780521679718
Author: Riley,Hobson,Bence
The third edition of this highly acclaimed undergraduate textbook is suitable for teaching all the mathematics for an undergraduate course in any of the physical sciences. As well as lucid descriptions of all the topics and many worked examples, it contains over 800 exercises. New stand-alone chapters give a systematic account of the ‘special functions’ of physical science, cover an extended range of practical applications of complex variables, and give an introduction to quantum operators. Further tabulations, of relevance in statistics and numerical integration, have been added. In this edition, half of the exercises are provided with hints and answers and, in a separate manual available to both students and their teachers, complete worked solutions.
Mathematical Methods for Physics and Engineering 3rd Table of contents:
1 Preliminary algebra
1.1 Simple functions and equations
1.1.1 Polynomials and polynomial equations
Example: the cubic case
A more general case
1.1.2 Factorising polynomials
1.1.3 Properties of roots
1.2 Trigonometric identities
1.2.1 Single-angle identities
1.2.2 Compound-angle identities
1.2.3 Double- and half-angle identities
1.3 Coordinate geometry
1.4 Partial fractions
1.4.1 Complications and special cases
The degree of the numerator is greater than or equal to that of the denominator
Factors of the form a + x in the denominator
Repeated factors in the denominator
1.5 Binomial expansion
1.5.1 Binomial coefficients
1.5.2 Proof of the binomial expansion
1.6 Properties of binomial coefficients
1.6.1 Identities involving binomial coefficients
1.6.2 Negative and non-integral values of n
1.7 Some particular methods of proof
1.7.1 Proof by induction
1.7.2 Proof by contradiction
1.7.3 Necessary and sufficient conditions
1.8 Exercises
Polynomial equations
Trigonometric identities
Coordinate geometry
Partial fractions
Binomial expansion
Proof by induction and contradiction
Necessary and sufficient conditions
1.9 Hints and answers
2 Preliminary calculus
2.1 Differentiation
2.1.1 Differentiation from first principles
2.1.2 Differentiation of products
2.1.3 The chain rule
2.1.4 Differentiation of quotients
2.1.5 Implicit differentiation
2.1.6 Logarithmic differentiation
2.1.7 Leibnitz’ theorem
2.1.8 Special points of a function
2.1.9 Curvature of a function
2.1.10 Theorems of differentiation
Rolle’s theorem
Mean value theorem
Applications of Rolle’s theorem and the mean value theorem
2.2 Integration
2.2.1 Integration from first principles
2.2.2 Integration as the inverse of differentiation
2.2.3 Integration by inspection
2.2.4 Integration of sinusoidal functions
2.2.5 Logarithmic integration
2.2.6 Integration using partial fractions
2.2.7 Integration by substitution
2.2.8 Integration by parts
2.2.9 Reduction formulae
2.2.10 Infinite and improper integrals
2.2.11 Integration in plane polar coordinates
2.2.12 Integral inequalities
2.2.13 Applications of integration
Mean value of a function
Finding the length of a curve
Surfaces of revolution
Volumes of revolution
2.3 Exercises
2.4 Hints and answers
3 Complex numbers and hyperbolic functions
3.1 The need for complex numbers
3.2 Manipulation of complex numbers
3.2.1 Addition and subtraction
3.2.2 Modulus and argument
3.2.3 Multiplication
3.2.4 Complex conjugate
3.2.5 Division
3.3 Polar representation of complex numbers
3.3.1 Multiplication and division in polar form
3.4 de Moivre’s theorem
3.4.1 Trigonometric identities
3.4.2 Finding the nth roots of unity
3.4.3 Solving polynomial equations
3.5 Complex logarithms and complex powers
3.6 Applications to differentiation and integration
3.7 Hyperbolic functions
3.7.1 Definitions
3.7.2 Hyperbolic–trigonometric analogies
3.7.3 Identities of hyperbolic functions
3.7.4 Solving hyperbolic equations
3.7.5 Inverses of hyperbolic functions
3.7.6 Calculus of hyperbolic functions
3.8 Exercises
3.9 Hints and answers
4 Series and limits
4.1 Series
4.2 Summation of series
4.2.1 Arithmetic series
4.2.2 Geometric series
4.2.3 Arithmetico-geometric series
4.2.4 The difference method
4.2.5 Series involving natural numbers
4.2.6 Transformation of series
4.3 Convergence of infinite series
4.3.1 Absolute and conditional convergence
4.3.2 Convergence of a series containing only real positive terms
Preliminary test
Comparison test
D’Alembert’s ratio test
Ratio comparison test
Quotient test
Integral test
Cauchy’s root test
Grouping terms
4.3.3 Alternating series test
4.4 Operations with series
4.5 Power series
4.5.1 Convergence of power series
4.5.2 Operations with power series
4.6 Taylor series
4.6.1 Taylor’s theorem
4.6.2 Approximation errors in Taylor series
4.6.3 Standard Maclaurin series
4.7 Evaluation of limits
Summary of methods for evaluating limits
4.8 Exercises
4.9 Hints and answers
5 Partial differentiation
5.1 Definition of the partial derivative
5.2 The total differential and total derivative
5.3 Exact and inexact differentials
5.4 Useful theorems of partial differentiation
5.5 The chain rule
5.6 Change of variables
5.7 Taylor’s theorem for many-variable functions
5.8 Stationary values of many-variable functions
5.9 Stationary values under constraints
5.10 Envelopes
5.10.1 Envelope equations
5.11 Thermodynamic relations
5.12 Differentiation of integrals
5.13 Exercises
5.14 Hints and answers
6 Multiple integrals
6.1 Double integrals
6.2 Triple integrals
6.3 Applications of multiple integrals
6.3.1 Areas and volumes
6.3.2 Masses, centres of mass and centroids
6.3.3 Pappus’ theorems
6.3.4 Moments of inertia
6.3.5 Mean values of functions
6.4 Change of variables in multiple integrals
6.4.1 Change of variables in double integrals
6.4.2 Evaluation of the integral I…
6.4.3 Change of variables in triple integrals
6.4.4 General properties of Jacobians
6.5 Exercises
6.6 Hints and answers
7 Vector algebra
7.1 Scalars and vectors
7.2 Addition and subtraction of vectors
7.3 Multiplication by a scalar
7.4 Basis vectors and components
7.5 Magnitude of a vector
7.6 Multiplication of vectors
7.6.1 Scalar product
7.6.2 Vector product
7.6.3 Scalar triple product
7.6.4 Vector triple product
7.7 Equations of lines, planes and spheres
7.7.1 Equation of a line
7.7.2 Equation of a plane
7.7.3 Equation of a sphere
7.8 Using vectors to find distances
7.8.1 Distance from a point to a line
7.8.2 Distance from a point to a plane
7.8.3 Distance from a line to a line
7.8.4 Distance from a line to a plane
7.9 Reciprocal vectors
7.10 Exercises
7.11 Hints and answers
8 Matrices and vector spaces
8.1 Vector spaces
8.1.1 Basis vectors
8.1.2 The inner product
8.1.3 Some useful inequalities
8.2 Linear operators
8.2.1 Properties of linear operators
8.3 Matrices
8.4 Basic matrix algebra
8.4.1 Matrix addition and multiplication by a scalar
8.4.2 Multiplication of matrices
8.4.3 The null and identity matrices
8.5 Functions of matrices
8.6 The transpose of a matrix
8.7 The complex and Hermitian conjugates of a matrix
8.8 The trace of a matrix
8.9 The determinant of a matrix
8.9.1 Properties of determinants
8.10 The inverse of a matrix
8.11 The rank of a matrix
8.12 Special types of square matrix
8.12.1 Diagonal matrices
8.12.2 Lower and upper triangular matrices
8.12.3 Symmetric and antisymmetric matrices
8.12.4 Orthogonal matrices
8.12.5 Hermitian and anti-Hermitian matrices
8.12.6 Unitary matrices
8.12.7 Normal matrices
8.13 Eigenvectors and eigenvalues
8.13.1 Eigenvectors and eigenvalues of a normal matrix
8.13.2 Eigenvectors and eigenvalues of Hermitian and anti-Hermitian matrices
8.13.3 Eigenvectors and eigenvalues of a unitary matrix
8.13.4 Eigenvectors and eigenvalues of a general square matrix
8.13.5 Simultaneous eigenvectors
8.14 Determination of eigenvalues and eigenvectors
8.14.1 Degenerate eigenvalues
8.15 Change of basis and similarity transformations
8.16 Diagonalisation of matrices
8.17 Quadratic and Hermitian forms
8.17.1 The stationary properties of the eigenvectors
8.17.2 Quadratic surfaces
8.18 Simultaneous linear equations
8.18.1 The range and null space of a matrix
No solution
A unique solution
Infinitely many solutions
8.18.2 N simultaneous linear equations in N unknowns
Direct inversion
LU decomposition
Cramer’s rule
8.18.3 Singular value decomposition
8.19 Exercises
8.20 Hints and answers
9 Normal modes
9.1 Typical oscillatory systems
9.2 Symmetry and normal modes
9.3 Rayleigh–Ritz method
9.4 Exercises
9.5 Hints and answers
10 Vector calculus
10.1 Differentiation of vectors
10.1.1 Differentiation of composite vector expressions
10.1.2 Differential of a vector
10.2 Integration of vectors
10.3 Space curves
10.4 Vector functions of several arguments
10.5 Surfaces
10.6 Scalar and vector fields
10.7 Vector operators
10.7.1 Gradient of a scalar field
10.7.2 Divergence of a vector field
10.7.3 Curl of a vector field
10.8 Vector operator formulae
10.8.1 Vector operators acting on sums and products
10.8.2 Combinations of grad, div and curl
10.9 Cylindrical and spherical polar coordinates
10.9.1 Cylindrical polar coordinates
10.9.2 Spherical polar coordinates
10.10 General curvilinear coordinates
Gradient
Divergence
Laplacian
Curl
10.11 Exercises
10.12 Hints and answers
11 Line, surface and volume integrals
11.1 Line integrals
11.1.1 Evaluating line integrals
11.1.2 Physical examples of line integrals
11.1.3 Line integrals with respect to a scalar
11.2 Connectivity of regions
11.3 Green’s theorem in a plane
11.4 Conservative fields and potentials
11.5 Surface integrals
11.5.1 Evaluating surface integrals
11.5.2 Vector areas of surfaces
11.5.3 Physical examples of surface integrals
11.6 Volume integrals
11.6.1 Volumes of three-dimensional regions
11.7 Integral forms for grad, div and curl
11.8 Divergence theorem and related theorems
11.8.1 Green’s theorems
11.8.2 Other related integral theorems
11.8.3 Physical applications of the divergence theorem
11.9 Stokes’ theorem and related theorems
11.9.1 Related integral theorems
11.9.2 Physical applications of Stokes’ theorem
11.10 Exercises
11.11 Hints and answers
12 Fourier series
12.1 The Dirichlet conditions
12.2 The Fourier coefficients
12.3 Symmetry considerations
12.4 Discontinuous functions
12.5 Non-periodic functions
12.6 Integration and differentiation
12.7 Complex Fourier series
12.8 Parseval’s theorem
12.9 Exercises
12.10 Hints and answers
13 Integral transforms
13.1 Fourier transforms
13.1.1 The uncertainty principle
13.1.2 Fraunhofer diffraction
13.1.3 The Dirac delta-function
13.1.4 Relation of the delta-function to Fourier transforms
13.1.5 Properties of Fourier transforms
13.1.6 Odd and even functions
13.1.7 Convolution and deconvolution
13.1.8 Correlation functions and energy spectra
13.1.9 Parseval’s theorem
13.1.10 Fourier transforms in higher dimensions
13.2 Laplace transforms
13.2.1 Laplace transforms of derivatives and integrals
13.2.2 Other properties of Laplace transforms
13.3 Concluding remarks
13.4 Exercises
13.5 Hints and answers
14 First-order ordinary differential equations
14.1 General form of solution
14.2 First-degree first-order equations
14.2.1 Separable-variable equations
14.2.2 Exact equations
14.2.3 Inexact equations: integrating factors
14.2.4 Linear equations
14.2.5 Homogeneous equations
14.2.6 Isobaric equations
14.2.7 Bernoulli’s equation
14.2.8 Miscellaneous equations
14.3 Higher-degree first-order equations
14.3.1 Equations soluble for p
14.3.2 Equations soluble for x
14.3.3 Equations soluble for y
14.3.4 Clairaut’s equation
14.4 Exercises
14.5 Hints and answers
15 Higher-order ordinary differential equations
15.1 Linear equations with constant coefficients
15.1.1 Finding the complementary function…
15.1.2 Finding the particular integral…
15.1.3 Constructing the general solution…
15.1.4 Linear recurrence relations
First-order recurrence relations
Second-order recurrence relations
Higher-order recurrence relations
15.1.5 Laplace transform method
15.2 Linear equations with variable coefficients
15.2.1 The Legendre and Euler linear equations
15.2.2 Exact equations
15.2.3 Partially known complementary function
15.2.4 Variation of parameters
15.2.5 Green’s functions
15.2.6 Canonical form for second-order equations
15.3 General ordinary differential equations
15.3.1 Dependent variable absent
15.3.2 Independent variable absent
15.3.3 Non-linear exact equations
15.3.4 Isobaric or homogeneous equations
15.3.5 Equations homogeneous in x or y alone
15.3.6 Equations having…as a solution
15.4 Exercises
15.5 Hints and answers
16 Series solutions of ordinary differential equations
16.1 Second-order linear ordinary differential equations
16.1.1 Ordinary and singular points of an ODE
16.2 Series solutions about an ordinary point
16.3 Series solutions about a regular singular point
16.3.1 Distinct roots not differing by an integer
16.3.2 Repeated root of the indicial equation
16.3.3 Distinct roots differing by an integer
16.4 Obtaining a second solution
16.4.1 The Wronskian method
16.4.2 The derivative method
16.4.3 Series form of the second solution
16.5 Polynomial solutions
16.6 Exercises
16.7 Hints and answers
17 Eigenfunction methods for differential equations
17.1 Sets of functions
17.1.1 Some useful inequalities
17.2 Adjoint, self-adjoint and Hermitian operators
17.3 Properties of Hermitian operators
17.3.1 Reality of the eigenvalues
17.3.2 Orthogonality and normalisation of the eigenfunctions
17.3.3 Completeness of the eigenfunctions
17.3.4 Construction of real eigenfunctions
17.4 Sturm–Liouville equations
17.4.1 Hermitian nature of the Sturm–Liouville operator
17.4.2 Transforming an equation into Sturm–Liouville form
17.5 Superposition of eigenfunctions: Green’s functions
17.6 A useful generalisation
17.7 Exercises
17.8 Hints and answers
18 Special functions
18.1 Legendre functions
18.1.1 Legendre functions for integer l
18.1.2 Properties of Legendre polynomials
Rodrigues’ formula
Mutual orthogonality
Generating function
Recurrence relations
18.2 Associated Legendre functions
18.2.1 Associated Legendre functions for integer l
18.2.2 Properties of associated Legendre functions P…
Mutual orthogonality
Generating function
Recurrence relations
18.3 Spherical harmonics
18.4 Chebyshev functions
18.4.1 Properties of Chebyshev polynomials
Rodrigues’ formula
Mutual orthogonality
Generating functions
Recurrence relations
18.5 Bessel functions
18.5.1 Bessel functions for non-integer v
18.5.2 Bessel functions for integer v
18.5.3 Properties of Bessel functions J…
Mutual orthogonality
Recurrence relations
Generating function
Integral representations
18.6 Spherical Bessel functions
18.7 Laguerre functions
18.7.1 Properties of Laguerre polynomials
Rodrigues’ formula
Mutual orthogonality
Generating function
Recurrence relations
18.8 Associated Laguerre functions
18.8.1 Properties of associated Laguerre polynomials
Rodrigues’ formula
Mutual orthogonality
Generating function
Recurrence relations
18.9 Hermite functions
18.9.1 Properties of Hermite polynomials
Rodrigues’ formula
Mutual orthogonality
Generating function
Recurrence relations
18.10 Hypergeometric functions
18.10.1 Properties of hypergeometric functions
Special cases
Integral representation
Relationships between hypergeometric functions
18.11 Confluent hypergeometric functions
18.11.1 Properties of confluent hypergeometric functions
Special cases
Integral representation
Relationships between confluent hypergeometric functions
18.12 The gamma function and related functions
18.12.1 The gamma function
18.12.2 The beta function
18.12.3 The incomplete gamma function
18.12.4 The error function
18.13 Exercises
18.14 Hints and answers
19 Quantum operators
19.1 Operator formalism
19.1.1 Commutation and commutators
19.2 Physical examples of operators
19.2.1 Angular momentum operators
Eigenvalues of the angular momentum operators
19.2.2 Uncertainty principles
19.2.3 Annihilation and creation operators
The energy spectrum of the simple harmonic oscillator
The normalisation of the eigenstates
19.3 Exercises
19.4 Hints and answers
20 Partial differential equations: general and particular solutions
20.1 Important partial differential equations
20.1.1 The wave equation
20.1.2 The diffusion equation
20.1.3 Laplace’s equation
20.1.4 Poisson’s equation
20.1.5 Schrodinger’s equation
20.2 General form of solution
20.3 General and particular solutions
20.3.1 First-order equations
20.3.2 Inhomogeneous equations and problems
20.3.3 Second-order equations
20.4 The wave equation
20.5 The diffusion equation
20.6 Characteristics and the existence of solutions
20.6.1 First-order equations
20.6.2 Second-order equations
20.7 Uniqueness of solutions
20.8 Exercises
20.9 Hints and answers
21 Partial differential equations: separation of variables and other methods
21.1 Separation of variables: the general method
21.2 Superposition of separated solutions
21.3 Separation of variables in polar coordinates
21.3.1 Laplace’s equation in polar coordinates
Laplace’s equation in plane polars
Laplace’s equation in cylindrical polars
Laplace’s equation in spherical polars
21.3.2 Other equations in polar coordinates
Helmholtz’s equation in plane polars
Helmholtz’s equation in cylindrical polars
Helmholtz’s equation in spherical polars
21.3.3 Solution by expansion
21.3.4 Separation of variables for inhomogeneous equations
21.4 Integral transform methods
21.5 Inhomogeneous problems – Green’s functions
21.5.1 Similarities to Green’s functions for ODEs
21.5.2 General boundary-value problems
21.5.3 Dirichlet problems
21.5.4 Neumann problems
21.6 Exercises
21.7 Hints and answers
22 Calculus of variations
22.1 The Euler–Lagrange equation
22.2 Special cases
22.2.1 F does not contain y explicitly
22.2.2 F does not contain x explicitly
22.3 Some extensions
22.3.1 Several dependent variables
22.3.2 Several independent variables
22.3.3 Higher-order derivatives
22.3.4 Variable end-points
22.4 Constrained variation
22.5 Physical variational principles
22.5.1 Fermat’s principle in optics
22.5.2 Hamilton’s principle in mechanics
22.6 General eigenvalue problems
22.7 Estimation of eigenvalues and eigenfunctions
22.8 Adjustment of parameters
22.9 Exercises
22.10 Hints and answers
23 Integral equations
23.1 Obtaining an integral equation from a differential equation
23.2 Types of integral equation
23.3 Operator notation and the existence of solutions
23.4 Closed-form solutions
23.4.1 Separable kernels
23.4.2 Integral transform methods
23.4.3 Differentiation
23.5 Neumann series
23.6 Fredholm theory
23.7 Schmidt–Hilbert theory
23.8 Exercises
23.9 Hints and answers
24 Complex variables
24.1 Functions of a complex variable
24.2 The Cauchy–Riemann relations
24.3 Power series in a complex variable
24.4 Some elementary functions
24.5 Multivalued functions and branch cuts
24.6 Singularities and zeros of complex functions
24.7 Conformal transformations
24.8 Complex integrals
24.9 Cauchy’s theorem
24.10 Cauchy’s integral formula
24.11 Taylor and Laurent series
24.12 Residue theorem
24.13 Definite integrals using contour integration
24.13.1 Integrals of sinusoidal functions
24.13.2 Some infinite integrals
24.13.3 Integrals of multivalued functions
24.14 Exercises
24.15 Hints and answers
25 Applications of complex variables
25.1 Complex potentials
25.2 Applications of conformal transformations
25.3 Location of zeros
25.4 Summation of series
25.5 Inverse Laplace transform
25.6 Stokes’ equation and Airy integrals
25.6.1 The solutions of Stokes’ equation
25.6.2 Series solution of Stokes’ equation
25.6.3 Contour integral solutions
25.7 WKB methods
25.7.1 Phase memory
25.7.2 Constructing the WKB solutions
25.7.3 Accuracy of the WKB solutions
25.7.4 The Stokes phenomenon
25.8 Approximations to integrals
25.8.1 Level lines and saddle points
25.8.2 Steepest descents method
25.8.3 Stationary phase method
25.9 Exercises
25.10 Hints and answers
26 Tensors
26.1 Some notation
26.2 Change of basis
26.3 Cartesian tensors
26.4 First- and zero-order Cartesian tensors
26.5 Second- and higher-order Cartesian tensors
26.6 The algebra of tensors
26.7 The quotient law
26.8 The tensors delta and epsilon
26.9 Isotropic tensors
26.10 Improper rotations and pseudotensors
26.11 Dual tensors
26.12 Physical applications of tensors
26.13 Integral theorems for tensors
26.14 Non-Cartesian coordinates
26.15 The metric tensor
26.16 General coordinate transformations and tensors
26.17 Relative tensors
26.18 Derivatives of basis vectors and Christoffel symbols
26.19 Covariant differentiation
26.20 Vector operators in tensor form
Gradient
Divergence
Laplacian
Curl
26.21 Absolute derivatives along curves
26.22 Geodesics
26.23 Exercises
26.24 Hints and answers
27 Numerical methods
27.1 Algebraic and transcendental equations
27.1.1 Rearrangement of the equation
27.1.2 Linear interpolation
27.1.3 Binary chopping
27.1.4 Newton–Raphson method
27.2 Convergence of iteration schemes
27.3 Simultaneous linear equations
27.3.1 Gaussian elimination
27.3.2 Gauss–Seidel iteration
27.3.3 Tridiagonal matrices
27.4 Numerical integration
27.4.1 Trapezium rule
27.4.2 Simpson’s rule
27.4.3 Gaussian integration
27.4.4 Monte Carlo methods
Crude Monte Carlo
Stratified sampling
Importance sampling
Control variates
Antithetic variates
Hit or miss method
Random number generation
27.5 Finite differences
27.6 Differential equations
27.6.1 Difference equations
27.6.2 Taylor series solutions
27.6.3 Prediction and correction
27.6.4 Runge–Kutta methods
27.6.5 Isoclines
27.7 Higher-order equations
27.8 Partial differential equations
27.9 Exercises
27.10 Hints and answers
28 Group theory
28.1 Groups
28.1.1 Definition of a group
28.1.2 Further examples of groups
28.2 Finite groups
28.3 Non-Abelian groups
28.4 Permutation groups
28.5 Mappings between groups
28.6 Subgroups
28.7 Subdividing a group
28.7.1 Equivalence relations and classes
28.7.2 Congruence and cosets
28.7.3 Conjugates and classes
28.8 Exercises
28.9 Hints and answers
29 Representation theory
29.1 Dipole moments of molecules
29.2 Choosing an appropriate formalism
29.3 Equivalent representations
29.4 Reducibility of a representation
29.5 The orthogonality theorem for irreducible representations
29.6 Characters
29.6.1 Orthogonality property of characters
29.7 Counting irreps using characters
29.7.1 Summation rules for irreps
29.8 Construction of a character table
29.9 Group nomenclature
29.10 Product representations
29.11 Physical applications of group theory
29.11.1 Bonding in molecules
29.11.2 Matrix elements in quantum mechanics
29.11.3 Degeneracy of normal modes
29.11.4 Breaking of degeneracies
29.12 Exercises
29.13 Hints and answers
30 Probability
30.1 Venn diagrams
30.2 Probability
30.2.1 Axioms and theorems
30.2.2 Conditional probability
30.2.3 Bayes’ theorem
30.3 Permutations and combinations
30.3.1 Permutations
30.3.2 Combinations
30.4 Random variables and distributions
30.4.1 Discrete random variables
30.4.2 Continuous random variables
30.4.3 Sets of random variables
30.5 Properties of distributions
30.5.1 Mean
30.5.2 Mode and median
30.5.3 Variance and standard deviation
30.5.4 Moments
30.5.5 Central moments
30.6 Functions of random variables
30.6.1 Discrete random variables
30.6.2 Continuous random variables
30.6.3 Functions of several random variables
30.6.4 Expectation values and variances
30.7 Generating functions
30.7.1 Probability generating functions
Sums of random variables
Variable-length sums of random variables
30.7.2 Moment generating functions
Scaling and shifting
Sums of random variables
Variable-length sums of random variables
Uniqueness
30.7.3 Characteristic function
30.7.4 Cumulant generating function
30.8 Important discrete distributions
30.8.1 The binomial distribution
The moment generating function for the binomial distribution
Multiple binomial distributions
30.8.2 The geometric and negative binomial distributions
30.8.3 The hypergeometric distribution
30.8.4 The Poisson distribution
The moment generating function for the Poisson distribution
The Poisson approximation to the binomial distribution
Multiple Poisson distributions
30.9 Important continuous distributions
30.9.1 The Gaussian distribution
The moment generating function for the Gaussian distribution
Gaussian approximation to the binomial distribution
Gaussian approximation to the Poisson distribution
Multiple Gaussian distributions
30.9.2 The log-normal distribution
30.9.3 The exponential and gamma distributions
30.9.4 The chi-squared distribution
30.9.5 The Cauchy and Breit–Wigner distributions
30.9.6 The uniform distribution
30.10 The central limit theorem
30.11 Joint distributions
30.11.1 Discrete bivariate distributions
30.11.2 Continuous bivariate distributions
30.11.3 Marginal and conditional distributions
30.12 Properties of joint distributions
30.12.1 Means
30.12.2 Variances
30.12.3 Covariance and correlation
30.13 Generating functions for joint distributions
30.14 Transformation of variables in joint distributions
30.15 Important joint distributions
30.15.1 The multinomial distribution
30.15.2 The multivariate Gaussian distribution
30.16 Exercises
30.17 Hints and answers
31 Statistics
31.1 Experiments, samples and populations
31.2 Sample statistics
31.2.1 Averages
31.2.2 Variance and standard deviation
31.2.3 Moments and central moments
31.2.4 Covariance and correlation
31.3 Estimators and sampling distributions
31.3.1 Consistency, bias and efficiency of estimators
Consistency
Bias
Efficiency
31.3.2 Fisher’s inequality
31.3.3 Standard errors on estimators
31.3.4 Confidence limits on estimators
31.3.5 Confidence limits for a Gaussian sampling distribution
31.3.6 Estimation of several quantities simultaneously
31.4 Some basic estimators
31.4.1 Population mean mu
31.4.2 Population variance sigma
31.4.3 Population standard deviation sigma
31.4.4 Population moments mu
31.4.5 Population central moments…
31.4.6 Population covariance Cov[x, y] and correlation Corr[x, y]
31.4.7 A worked example
31.5 Maximum-likelihood method
31.5.1 The maximum-likelihood estimator
31.5.2 Transformation invariance and bias of ML estimators
31.5.3 Efficiency of ML estimators
31.5.4 Standard errors and confidence limits on ML estimators
31.5.5 The Bayesian interpretation of the likelihood function
31.5.6 Behaviour of ML estimators for large N
31.5.7 Extended maximum-likelihood method
31.6 The method of least squares
31.6.1 Linear least squares
31.6.2 Non-linear least squares
31.7 Hypothesis testing
31.7.1 Simple and composite hypotheses
31.7.2 Statistical tests
31.7.3 The Neyman–Pearson test
31.7.4 The generalised likelihood-ratio test
31.7.5 Student’s t-test
31.7.6 Fisher’s F-test
31.7.7 Goodness of fit in least-squares problems
31.8 Exercises
31.9 Hints and answers
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