Table of Contents Standard Mathematical Tables and Formulae (31^st edition)
(by Zwillinger)

• Editor-in-Chief
• Associate Editors
• Editorial Advisory Board
• Preface
• Contributors

Chapter   1      Analysis

• 1.1   Constants
  • 1.1.1   Types of numbers
  • 1.1.2   Roman numerals
  • 1.1.3   Arrow notation
  • 1.1.4   Representation of numbers
  • 1.1.5   Binary prefixes
  • 1.1.6   Decimal multiples and prefixes
  • 1.1.7   Decimal equivalents of common fractions
  • 1.1.8   Hexadecimal addition and subtraction table
  • 1.1.9   Hexadecimal multiplication table
  • 1.1.10   Hexadecimal--decimal fraction conversion table
• 1.2   Special numbers
  • 1.2.1   Powers of 2
  • 1.2.2   Powers of 16 in decimal scale
  • 1.2.3   Powers of 10 in hexadecimal scale
  • 1.2.4   Special constants
  • 1.2.5   Constants in different bases
  • 1.2.6   Factorials
  • 1.2.7   Bernoulli polynomials and numbers
  • 1.2.8   Euler polynomials and numbers
  • 1.2.9   Fibonacci numbers
  • 1.2.10   Powers of integers
  • 1.2.11   Sums of powers of integers
  • 1.2.12   Negative integer powers
  • 1.2.13   de-Bruijn sequences
  • 1.2.14   Integer sequences
• 1.3   Series and products
  • 1.3.1   Definitions
  • 1.3.2   General properties
  • 1.3.3   Convergence tests
  • 1.3.4   Types of series
  • 1.3.5   Summation formulae
  • 1.3.6   Improving convergence: Shanks transformation
  • 1.3.7   Summability methods
  • 1.3.8   Operations with power series
  • 1.3.9   Miscellaneous sums and series
  • 1.3.10   Infinite series
  • 1.3.11   Infinite products
  • 1.3.12   Infinite products and infinite series
• 1.4   Fourier series
  • 1.4.1   Special cases
  • 1.4.2   Alternate forms
  • 1.4.3   Useful series
  • 1.4.4   Expansions of basic periodic functions
• 1.5   Complex analysis
  • 1.5.1   Definitions
  • 1.5.2   Operations on complex numbers
  • 1.5.3   Functions of a complex variable
  • 1.5.4   Cauchy--Riemann equations
  • 1.5.5   Cauchy integral theorem
  • 1.5.6   Cauchy integral formula
  • 1.5.7   Taylor series expansions
  • 1.5.8   Laurent series expansions
  • 1.5.9   Zeros and singularities
  • 1.5.10   Residues
  • 1.5.11   The argument principle
  • 1.5.12   Transformations and mappings
• 1.6   Interval analysis
  • 1.6.1   Interval arithmetic rules
  • 1.6.2   Interval arithmetic properties
• 1.7   Real analysis
  • 1.7.1   Relations
  • 1.7.2   Functions (mappings)
  • 1.7.3   Sets of real numbers
  • 1.7.4   Topological space
  • 1.7.5   Metric space
  • 1.7.6   Convergence in R with metric |x-y|
  • 1.7.7   Continuity in R with metric |x-y|
  • 1.7.8   Banach space
  • 1.7.9   Hilbert space
  • 1.7.10   Asymptotic relationships
• 1.8   Generalized Functions
  • 1.8.1   Delta function
  • 1.8.2   Other generalized functions

Chapter   2      Algebra

• 2.1   Proofs without words
• 2.2   Elementary algebra
  • 2.2.1   Basic algebra
  • 2.2.2   Progressions
  • 2.2.3   DeMoivre's theorem
  • 2.2.4   Partial fractions
• 2.3   Polynomials
  • 2.3.1   Quadratic polynomials
  • 2.3.2   Cubic polynomials
  • 2.3.3   Quartic polynomials
  • 2.3.4   Quartic curves
  • 2.3.5   Quintic polynomials
  • 2.3.6   Tschirnhaus' transformation
  • 2.3.7   Polynomial norms
  • 2.3.8   Cyclotomic polynomials
  • 2.3.9   Other polynomial properties
• 2.4   Number theory
  • 2.4.1   Divisibility
  • 2.4.2   Congruences
  • 2.4.3   Chinese remainder theorem
  • 2.4.4   Continued fractions
  • 2.4.5   Diophantine equations
  • 2.4.6   Greatest common divisor
  • 2.4.7   Least common multiple
  • 2.4.8   Magic squares
  • 2.4.9   Mobius function
  • 2.4.10   Prime numbers
  • 2.4.11   Prime numbers of special forms
  • 2.4.12   Prime numbers less than 100,000
  • 2.4.13   Factorization table
  • 2.4.14   Factorization of 2^m-1
  • 2.4.15   Euler Totient function
• 2.5   Vector algebra
  • 2.5.1   Notation for vectors and scalars
  • 2.5.2   Physical vectors
  • 2.5.3   Fundamental definitions
  • 2.5.4   Laws of vector algebra
  • 2.5.5   Vector norms
  • 2.5.6   Dot, scalar, or inner product
  • 2.5.7   Vector or cross product
  • 2.5.8   Scalar and vector triple products
• 2.6   Linear and matrix algebra
  • 2.6.1   Definitions
  • 2.6.2   Types of matrices
  • 2.6.3   Conformability for addition and multiplication
  • 2.6.4   Determinants and permanents
  • 2.6.5   Matrix norms
  • 2.6.6   Singularity, rank, and inverses
  • 2.6.7   Systems of linear equations
  • 2.6.8   Linear spaces and linear mappings
  • 2.6.9   Traces
  • 2.6.10   Generalized inverses
  • 2.6.11   Eigenstructure
  • 2.6.12   Matrix diagonalization
  • 2.6.13   Matrix exponentials
  • 2.6.14   Quadratic forms
  • 2.6.15   Matrix factorizations
  • 2.6.16   Theorems
  • 2.6.17   The vector operation
  • 2.6.18   Kronecker products
  • 2.6.19   Kronecker sums
• 2.7   Abstract algebra
  • 2.7.1   Definitions
  • 2.7.2   Groups
  • 2.7.3   Rings
  • 2.7.4   Fields
  • 2.7.5   Quadratic fields
  • 2.7.6   Finite fields
  • 2.7.7   Homomorphisms and isomorphisms
  • 2.7.8   Matrix classes that are groups
  • 2.7.9   Permutation groups
  • 2.7.10   Tables

Chapter   3      Discrete Mathematics

• 3.1   Symbolic logic
  • 3.1.1   Propositional calculus
  • 3.1.2   Tautologies
  • 3.1.3   Truth tables as functions
  • 3.1.4   Rules of inference
  • 3.1.5   Deductions
  • 3.1.6   Predicate calculus
• 3.2   Set theory
  • 3.2.1   Sets
  • 3.2.2   Set operations and relations
  • 3.2.3   Connection between sets and probability
  • 3.2.4   Venn diagrams
  • 3.2.5   Paradoxes and theorems of set theory
  • 3.2.6   Inclusion/Exclusion
  • 3.2.7   Partially ordered sets
• 3.3   Combinatorics
  • 3.3.1   Sample selection
  • 3.3.2   Balls into cells
  • 3.3.3   Binomial coefficients
  • 3.3.4   Multinomial coefficients
  • 3.3.5   Arrangements and derangements
  • 3.3.6   Partitions
  • 3.3.7   Bell numbers
  • 3.3.8   Catalan numbers
  • 3.3.9   Stirling cycle numbers
  • 3.3.10   Stirling subset numbers
  • 3.3.11   Tables
• 3.4   Graphs
  • 3.4.1   Notation
  • 3.4.2   Basic definitions
  • 3.4.3   Constructions
  • 3.4.4   Fundamental results
  • 3.4.5   Tree diagrams
• 3.5   Combinatorial design theory
  • 3.5.1   t-Designs
  • 3.5.2   Balanced incomplete block designs (BIBDs)
  • 3.5.3   Difference sets
  • 3.5.4   Finite geometry
  • 3.5.5   Steiner triple systems
  • 3.5.6   Hadamard matrices
  • 3.5.7   Latin squares
  • 3.5.8   Room squares
  • 3.5.9   Costas arrays
• 3.6   Communication theory
  • 3.6.1   Information theory
  • 3.6.2   Block coding
  • 3.6.3   Source coding for English text
  • 3.6.4   Morse code
  • 3.6.5   Gray code
  • 3.6.6   Finite fields
  • 3.6.7   Binary sequences
• 3.7   Difference equations
  • 3.7.1   The calculus of finite differences
  • 3.7.2   Existence and uniqueness
  • 3.7.3   Linear independence: general solution
  • 3.7.4   Homogeneous equations with constant coefficients
  • 3.7.5   Non-homogeneous equations
  • 3.7.6   Generating functions and Z transforms
  • 3.7.7   Closed-form solutions for special equations
• 3.8   Discrete Dynamical Systems and Chaos
  • 3.8.1   Chaotic one-dimensional maps
  • 3.8.2   Logistic map
  • 3.8.3   Julia sets and the Mandelbrot set
• 3.9   Game theory
  • 3.9.1   Two person non-cooperative matrix games
  • 3.9.2   Voting power
• 3.10   Operations research
  • 3.10.1   Linear programming
  • 3.10.2   Duality and complementary slackness
  • 3.10.3   Linear integer programming
  • 3.10.4   Branch and bound
  • 3.10.5   Network flow methods
  • 3.10.6   Assignment problem
  • 3.10.7   Dynamic programming
  • 3.10.8   Shortest path problem
  • 3.10.9   Heuristic search techniques

Chapter   4      Geometry

• 4.1   Coordinate systems in the plane
  • 4.1.1   Convention
  • 4.1.2   Substitutions and transformations
  • 4.1.3   Cartesian coordinates in the plane
  • 4.1.4   Polar coordinates in the plane
  • 4.1.5   Homogeneous coordinates in the plane
  • 4.1.6   Oblique coordinates in the plane
• 4.2   Plane symmetries or isometries
  • 4.2.1   Formulae for symmetries: Cartesian coordinates
  • 4.2.2   Formulae for symmetries: homogeneous coordinates
  • 4.2.3   Formulae for symmetries: polar coordinates
  • 4.2.4   Crystallographic groups
  • 4.2.5   Classifying the crystallographic groups
• 4.3   Other transformations of the plane
  • 4.3.1   Similarities
  • 4.3.2   Affine transformations
  • 4.3.3   Projective transformations
• 4.4   Lines
  • 4.4.1   Lines with prescribed properties
  • 4.4.2   Distances
  • 4.4.3   Angles
  • 4.4.4   Concurrence and collinearity
• 4.5   Polygons
  • 4.5.1   Triangles
  • 4.5.2   Quadrilaterals
  • 4.5.3   Regular polygons
• 4.6   Conics
  • 4.6.1   Alternative characterization
  • 4.6.2   The general quadratic equation
  • 4.6.3   Additional properties of ellipses
  • 4.6.4   Additional properties of hyperbolas
  • 4.6.5   Additional properties of parabolas
  • 4.6.6   Circles
• 4.7   Special plane curves
  • 4.7.1   Algebraic curves
  • 4.7.2   Roulettes (spirograph curves)
  • 4.7.3   Curves in polar coordinates
  • 4.7.4   Spirals
  • 4.7.5   The Peano curve and fractal curves
  • 4.7.6   Fractal objects
  • 4.7.7   Classical constructions
• 4.8   Coordinate systems in space
  • 4.8.1   Cartesian coordinates in space
  • 4.8.2   Cylindrical coordinates in space
  • 4.8.3   Spherical coordinates in space
  • 4.8.4   Relations between Cartesian, cylindrical, and spherical coordinates
  • 4.8.5   Homogeneous coordinates in space
• 4.9   Space symmetries or isometries
  • 4.9.1   Formulae for symmetries: Cartesian coordinates
  • 4.9.2   Formulae for symmetries: homogeneous coordinates
• 4.10   Other transformations of space
  • 4.10.1   Similarities
  • 4.10.2   Affine transformations
  • 4.10.3   Projective transformations
• 4.11   Direction Angles and Direction Cosines
• 4.12   Planes
  • 4.12.1   Planes with prescribed properties
  • 4.12.2   Concurrence and coplanarity
• 4.13   Lines in space
  • 4.13.1   Distances
  • 4.13.2   Angles
  • 4.13.3   Concurrence, coplanarity, parallelism
• 4.14   Polyhedra
  • 4.14.1   Convex regular polyhedra
  • 4.14.2   Polyhedra nets
• 4.15   Cylinders
• 4.16   Cones
• 4.17   Surfaces of revolution: the torus
• 4.18   Quadrics
  • 4.18.1   Spheres
• 4.19   Spherical geometry & trigonometry
  • 4.19.1   Right spherical triangles
  • 4.19.2   Oblique spherical triangles
• 4.20   Differential geometry
  • 4.20.1   Curves
  • 4.20.2   Surfaces
• 4.21   Angle conversion
• 4.22   Knots up to eight crossings

Chapter   5      Continuous Mathematics

• 5.1   Differential calculus
  • 5.1.1   Limits
  • 5.1.2   Derivatives
  • 5.1.3   Derivatives of common functions
  • 5.1.4   Derivative formulae
  • 5.1.5   Derivative theorems
  • 5.1.6   The two-dimensional chain rule
  • 5.1.7   l'Hospital's rule
  • 5.1.8   Maxima and minima of functions
  • 5.1.9   Vector calculus
  • 5.1.10   Matrix and vector derivatives
• 5.2   Differential forms
  • 5.2.1   Products of 1-forms
  • 5.2.2   Differential 2-forms
  • 5.2.3   The 2-forms in Rⁿ
  • 5.2.4   Higher dimensional forms
  • 5.2.5   The exterior derivative
  • 5.2.6   Properties of the exterior derivative
• 5.3   Integration
  • 5.3.1   Definitions
  • 5.3.2   Properties of integrals
  • 5.3.3   Methods of evaluating integrals
  • 5.3.4   Types of integrals
  • 5.3.5   Integral inequalities
  • 5.3.6   Convergence tests
  • 5.3.7   Variational principles
  • 5.3.8   Continuity of integral anti-derivatives
  • 5.3.9   Asymptotic integral evaluation
  • 5.3.10   Special functions defined by integrals
  • 5.3.11   Applications of integration
  • 5.3.12   Moments of inertia for various bodies
  • 5.3.13   Tables of integrals
• 5.4   Table of Indefinite Integrals
  • 5.4.1   Elementary forms
  • 5.4.2   Forms containing a+bx
  • 5.4.3   Forms containing c² +- x² and x²-c²
  • 5.4.4   Forms containing a+bx and c+dx
  • 5.4.5   Forms containing a+bxⁿ
  • 5.4.6   Forms containing c³ +- x³
  • 5.4.7   Forms containing c⁴ +- x⁴
  • 5.4.8   Forms containing a+b x+c x²
  • 5.4.9   Forms containing SQRT(a+b x  
  • 5.4.10   Forms containing SQRT(a+b x) and SQRT(c+d x)
  • 5.4.11   Forms containing SQRT(x² +- a²)
  • 5.4.12   Forms containing SQRT(a²-x²)
  • 5.4.13   Forms containing SQRT(a+bx+cx²)
  • 5.4.14   Forms containing SQRT(2ax-x²)
  • 5.4.15   Miscellaneous algebraic forms
  • 5.4.16   Forms involving trigonometric functions
  • 5.4.17   Forms involving inverse trigonometric functions
  • 5.4.18   Logarithmic forms
  • 5.4.19   Exponential forms
  • 5.4.20   Hyperbolic forms
  • 5.4.21   Bessel functions
• 5.5   Table of definite integrals
  • 5.5.1   Table of semi-integrals
• 5.6   Ordinary differential equations
  • 5.6.1   Linear differential equations
  • 5.6.2   Solution techniques
  • 5.6.3   Integrating factors
  • 5.6.4   Variation of parameters
  • 5.6.5   Green's functions
  • 5.6.6   Table of Green's functions
  • 5.6.7   Transform techniques
  • 5.6.8   Named ordinary differential equations
  • 5.6.9   Liapunov's direct method
  • 5.6.10   Lie groups
  • 5.6.11   Stochastic differential equations
  • 5.6.12   Types of critical points
• 5.7   Partial differential equations
  • 5.7.1   Classifications of PDEs
  • 5.7.2   Named partial differential equations
  • 5.7.3   Transforming partial differential equations
  • 5.7.4   Well-posedness of PDEs
  • 5.7.5   Green's functions
  • 5.7.6   Quasi-linear equations
  • 5.7.7   Separation of variables
  • 5.7.8   Solutions of Laplace's equation
  • 5.7.9   Solutions to the wave equation
  • 5.7.10   Particular solutions to some PDEs
• 5.8   Eigenvalues
• 5.9   Integral equations
  • 5.9.1   Definitions
  • 5.9.2   Connection to differential equations
  • 5.9.3   Fredholm alternative
  • 5.9.4   Special equations with solutions
• 5.10   Tensor analysis
  • 5.10.1   Definitions
  • 5.10.2   Algebraic tensor operations
  • 5.10.3   Differentiation of tensors
  • 5.10.4   Metric tensor
  • 5.10.5   Results
  • 5.10.6   Examples of tensors
• 5.11   Orthogonal coordinate systems
  • 5.11.1   Table of orthogonal coordinate systems
• 5.12   Control theory

Chapter   6      Special Functions

• 6.1   Trigonometric or circular functions
  • 6.1.1   Definition of angles
  • 6.1.2   Characterization of angles
  • 6.1.3   Circular functions
  • 6.1.4   Circular functions of special angles
  • 6.1.5   Evaluating sines and cosines at multiples of pi
  • 6.1.6   Symmetry and periodicity relationships
  • 6.1.7   Functions in terms of angles in the first quadrant
  • 6.1.8   One circular function in terms of another
  • 6.1.9   Circular functions in terms of exponentials
  • 6.1.10   Fundamental identities
  • 6.1.11   Angle sum and difference relationships
  • 6.1.12   Double angle formulae
  • 6.1.13   Multiple angle formulae
  • 6.1.14   Half angle formulae
  • 6.1.15   Powers of circular functions
  • 6.1.16   Products of sine and cosine
  • 6.1.17   Sums of circular functions
• 6.2   Circular functions and planar triangles
  • 6.2.1   Right triangles
  • 6.2.2   General plane triangles
  • 6.2.3   Half angle formulae
  • 6.2.4   Solution of triangles
  • 6.2.5   Tables of trigonometric functions
• 6.3   Inverse circular functions
  • 6.3.1   Definition in terms of an integral
  • 6.3.2   Principal values of the inverse circular functions
  • 6.3.3   Fundamental identities
  • 6.3.4   Functions of negative arguments
  • 6.3.5   Relationship to inverse hyperbolic functions
  • 6.3.6   Sum and difference of two inverse trigonometric functions
• 6.4   Ceiling and floor functions
• 6.5   Exponential function
  • 6.5.1   Exponentiation
  • 6.5.2   Definition of e^z
  • 6.5.3   Derivative and integral of e^x
• 6.6   Logarithmic functions
  • 6.6.1   Definition of the natural logarithm
  • 6.6.2   Logarithm of special values
  • 6.6.3   Relating the logarithm to the exponential
  • 6.6.4   Identities
  • 6.6.5   Series expansions for the natural logarithm
  • 6.6.6   Derivative and integration formulae
• 6.7   Hyperbolic functions
  • 6.7.1   Definitions of the hyperbolic functions
  • 6.7.2   Range of values
  • 6.7.3   Hyperbolic functions in terms of one another
  • 6.7.4   Relations among hyperbolic functions
  • 6.7.5   Relationship to circular functions
  • 6.7.6   Series expansions
  • 6.7.7   Symmetry relationships
  • 6.7.8   Sum and difference formulae
  • 6.7.9   Multiple argument relations
  • 6.7.10   Sums of functions
  • 6.7.11   Products of functions
  • 6.7.12   Half--argument formulae
  • 6.7.13   Differentiation formulae
• 6.8   Inverse hyperbolic functions
  • 6.8.1   Range of values
  • 6.8.2   Relationships among inverse hyperbolic functions
  • 6.8.3   Relationships with logarithmic functions
  • 6.8.4   Relationships with circular functions
  • 6.8.5   Sum and difference of functions
• 6.9   Gudermannian Function
  • 6.9.1   Fundamental identities
  • 6.9.2   Derivatives of Gudermannian
  • 6.9.3   Relationship to hyperbolic and circular functions
  • 6.9.4   Numerical values of hyperbolic functions
• 6.10   Orthogonal polynomials
  • 6.10.1   Hermite polynomials
  • 6.10.2   Jacobi polynomials
  • 6.10.3   Laguerre polynomials
  • 6.10.4   Generalized Laguerre polynomials
  • 6.10.5   Legendre polynomials
  • 6.10.6   Chebyshev polynomials, first kind
  • 6.10.7   Chebyshev polynomials, second kind
  • 6.10.8   Tables of orthogonal polynomials
  • 6.10.9   Zernike polynomials
  • 6.10.10   Spherical harmonics
• 6.11   Gamma function
  • 6.11.1   Recursion formula
  • 6.11.2   Gamma function of special values
  • 6.11.3   Properties
  • 6.11.4   Asymptotic expansion
  • 6.11.5   Logarithmic derivative of the gamma function
  • 6.11.6   Numerical values
• 6.12   Beta function
  • 6.12.1   Numerical values of the beta function
• 6.13   Error functions
  • 6.13.1   Properties
  • 6.13.2   Error function of special values
  • 6.13.3   Expansions
  • 6.13.4   Special cases
• 6.14   Fresnel integrals
  • 6.14.1   Properties
  • 6.14.2   Asymptotic expansion
  • 6.14.3   Numerical values of error functions and Fresnel integrals
• 6.15   Sine, cosine, and exponential integrals
  • 6.15.1   Sine and cosine integrals
  • 6.15.2   Exponential integrals
  • 6.15.3   Logarithmic integral
  • 6.15.4   Numerical values
• 6.16   Polylogarithms
  • 6.16.1   Polylogarithms of special values
  • 6.16.2   Polylogarithms properties
• 6.17   Hypergeometric functions
  • 6.17.1   Special cases
  • 6.17.2   Properties
  • 6.17.3   Recursion formulae
• 6.18   Legendre functions
  • 6.18.1   Differential equation: Legendre function
  • 6.18.2   Definition
  • 6.18.3   Singular points
  • 6.18.4   Relationships
  • 6.18.5   Recursion relationships
  • 6.18.6   Integrals
  • 6.18.7   Polynomial case
  • 6.18.8   Differential equation: associated Legendre function
  • 6.18.9   Relationships between the associated and ordinary Legendre functions
  • 6.18.10   Orthogonality relationship
  • 6.18.11   Recursion relationships
• 6.19   Bessel functions
  • 6.19.1   Differential equation
  • 6.19.2   Singular points
  • 6.19.3   Relationships
  • 6.19.4   Series expansions
  • 6.19.5   Recurrence relationships
  • 6.19.6   Behavior as z->0
  • 6.19.7   Integrals
  • 6.19.8   Fourier expansion
  • 6.19.9   Auxiliary functions
  • 6.19.10   Inverse relationships
  • 6.19.11   Asymptotic expansions
  • 6.19.12   Zeros of Bessel functions
  • 6.19.13   Half order Bessel functions
  • 6.19.14   Modified Bessel functions
  • 6.19.15   Airy functions
  • 6.19.16   Numerical values for the Bessel functions
• 6.20   Elliptic integrals
  • 6.20.1   Definitions
  • 6.20.2   Properties
  • 6.20.3   Numerical values of the elliptic integrals
• 6.21   Jacobian elliptic functions
  • 6.21.1   Properties
  • 6.21.2   Derivatives and integrals
  • 6.21.3   Series expansions
• 6.22   Clebsch--Gordan coefficients
• 6.23   Integral transforms: Preliminaries
• 6.24   Fourier integral transform
  • 6.24.1   Existence
  • 6.24.2   Properties
  • 6.24.3   Inversion formula
  • 6.24.4   Poisson summation formula
  • 6.24.5   Shannon's sampling theorem
  • 6.24.6   Uncertainty principle
  • 6.24.7   Fourier sine and cosine transforms
• 6.25   Discrete Fourier transform (DFT)
  • 6.25.1   Properties
• 6.26   Fast Fourier transform (FFT)
• 6.27   Multidimensional Fourier transforms
• 6.28   Laplace transform
  • 6.28.1   Existence and domain of convergence
  • 6.28.2   Properties
  • 6.28.3   Inversion formulae
  • 6.28.4   Convolution
• 6.29   Hankel transform
  • 6.29.1   Properties
• 6.30   Hartley transform
• 6.31   Hilbert transform
  • 6.31.1   Existence
  • 6.31.2   Properties
  • 6.31.3   Relationship with the Fourier transform
• 6.32   Z-Transform
  • 6.32.1   Examples
  • 6.32.2   Properties
  • 6.32.3   Inversion formula
  • 6.32.4   Convolution and product
• 6.33   Tables of transforms

Chapter   7      Probability and Statistics

• 7.1   Probability theory
  • 7.1.1   Introduction
  • 7.1.2   Multivariate distributions
  • 7.1.3   Random sums of random variables
  • 7.1.4   Transforming variables
  • 7.1.5   Central limit theorem
  • 7.1.6   Inequalities
  • 7.1.7   Averages over vectors
  • 7.1.8   Geometric probability
• 7.2   Classical probability problems
  • 7.2.1   Raisin cookie problem
  • 7.2.2   Gambler's ruin problem
  • 7.2.3   Card games
  • 7.2.4   Distribution of dice sums
  • 7.2.5   Birthday problem
• 7.3   Probability distributions
  • 7.3.1   Discrete distributions
  • 7.3.2   Continuous distributions
• 7.4   Queuing theory
  • 7.4.1   Variables
  • 7.4.2   Theorems
• 7.5   Markov chains
  • 7.5.1   Transition function and matrix
  • 7.5.2   Recurrence
  • 7.5.3   Stationary distributions
  • 7.5.4   Random walks
  • 7.5.5   Ehrenfest chain
• 7.6   Random number generation
  • 7.6.1   Methods of pseudorandom number generation
  • 7.6.2   Generating non-uniform random variables
• 7.7   Control charts and reliability
  • 7.7.1   Control charts
  • 7.7.2   Acceptance sampling
  • 7.7.3   Reliability
  • 7.7.4   Failure time distributions
• 7.8   Risk analysis and decision rules
• 7.9   Statistics
  • 7.9.1   Descriptive statistics
  • 7.9.2   Statistical estimators
  • 7.9.3   Cramer--Rao bound
  • 7.9.4   Order statistics
  • 7.9.5   Classic statistics problems
• 7.10   Confidence intervals
  • 7.10.1   Confidence interval: sample from one population
  • 7.10.2   Confidence interval: samples from two populations
• 7.11   Tests of hypotheses
  • 7.11.1   Hypothesis tests: parameter from one population
  • 7.11.2   Hypothesis tests: parameters from two populations
  • 7.11.3   Hypothesis tests: distribution of a population
  • 7.11.4   Hypothesis tests: distributions of two populations
  • 7.11.5   Sequential probability ratio tests
• 7.12   Linear regression
  • 7.12.1   Linear model y_i=b₀+b₁x_i+e
  • 7.12.2   General model y=b₀+b₁x₁+b₂x₂+...+b_nx_n+e
• 7.13   Analysis of variance (ANOVA)
  • 7.13.1   One-factor ANOVA
  • 7.13.2   Unreplicated two-factor ANOVA
  • 7.13.3   Replicated two-factor ANOVA
• 7.14   Probability tables
  • 7.14.1   Critical values
  • 7.14.2   Table of the normal distribution
  • 7.14.3   Percentage points, Student's t-distribution
  • 7.14.4   Percentage points, chi-square distribution
  • 7.14.5   Percentage points, F-distribution
  • 7.14.6   Cumulative terms, binomial distribution
  • 7.14.7   Cumulative terms, Poisson distribution
  • 7.14.8   Critical values, Kolmogorov--Smirnov test
  • 7.14.9   Critical values, two sample Kolmogorov--Smirnov test
  • 7.14.10   Critical values, Spearman's rank correlation
• 7.15   Signal processing
  • 7.15.1   Estimation
  • 7.15.2   Kalman filters
  • 7.15.3   Matched filtering (Wiener filter)
  • 7.15.4   Walsh functions
  • 7.15.5   Wavelets

Chapter   8      Scientific Computing

• 8.1   Basic numerical analysis
  • 8.1.1   Approximations and errors
  • 8.1.2   Solution to algebraic equations
  • 8.1.3   Interpolation
  • 8.1.4   Fitting equations to data
• 8.2   Numerical linear algebra
  • 8.2.1   Solving linear systems
  • 8.2.2   Gaussian elimination
  • 8.2.3   Gaussian elimination algorithm
  • 8.2.4   Pivoting
  • 8.2.5   Eigenvalue computation
  • 8.2.6   Householder's method
  • 8.2.7   QR algorithm
  • 8.2.8   Non-linear systems and numerical optimization
• 8.3   Numerical integration and differentiation
  • 8.3.1   Numerical integration
  • 8.3.2   Numerical differentiation
  • 8.3.3   Numerical summation
• 8.4   Programming techniques

Chapter   9      Financial Analysis

• 9.1   Financial formulae
  • 9.1.1   Definition of financial terms
  • 9.1.2   Formulae connecting financial terms
  • 9.1.3   Examples
• 9.2   Financial tables
  • 9.2.1   Compound interest: find final value
  • 9.2.2   Compound interest: find interest rate
  • 9.2.3   Compound interest: find annuity

Chapter   10      Miscellaneous

• 10.1   Units
  • 10.1.1   SI system of measurement
  • 10.1.2   United States customary system of weights and measures
  • 10.1.3   Physical constants
  • 10.1.4   Dimensional analysis/Buckingham pi
  • 10.1.5   Units of physical quantities
  • 10.1.6   Conversion: metric to English
  • 10.1.7   Conversion: English to metric
  • 10.1.8   Miscellaneous conversions
  • 10.1.9   Temperature conversion
• 10.2   Interpretations of powers of 10
• 10.3   Calendar computations
  • 10.3.1   Leap years
  • 10.3.2   Day of week for any given day
  • 10.3.3   Number of each day of the year
• 10.4   AMS classification scheme
• 10.5   Fields medals
• 10.6   Greek alphabet
• 10.7   Computer languages
  • 10.7.1   Software contact information
• 10.8   Professional Mathematical Organizations
• 10.9   Electronic mathematical resources
• 10.10   Biographies of mathematicians