ANALYSIS
Constants
Types of Numbers. Representation of Numbers. Decimal Multiples and Prefixes. Roman Numerals. Decimal Equivalents of Common Fractions. Hexadecimal Addition and Subtraction Table. Hexadecimal Multiplication Table. Hexadecimal-Decimal Integer Conversion Table.
Special Numbers
Positive Powers of 2. Negative Powers of 2. Powers of 16 in Decimal Scale. Powers of 10 in Hexadecimal Scale. Special Constants. Factorials. Important Numbers in Different Bases. Bernoulli Numbers and Polynomials. Euler Numbers and Polynomials. Fibonacci Numbers. Powers of Integers. Sums of Powers of Integers. Negative Integer Powers. Integer Sequences. de Bruijn Sequences.
Series
Definitions. General Properties. Convergence Tests. Types of Series. Summation Formulae. Improving Convergence: Shanks Transformation. Summability Methods. Operations with Series. Miscellaneous Sums and Series. Infinite Series. Infinite Products.
Fourier Series.
Special Cases. Alternate Forms. Expansions of Basic Periodic Functions.
Complex Analysis
Definitions. Operations on Complex Numbers. Powers and Roots of Complex Numbers. Functions of a Complex Variable. Cauchy-Riemann Equations. Cauchy Integral Theorem. Cauchy Integral Formula. Taylor Series Expansions. Laurent Series Expansions. Zeros and Singularities. Residues. Principle of the Argument. Transformations and Mappings. Bilinear Transformations. Table of Transformations. Table of Conformal Mapping.
Real Analysis
Relations. Functions (Mappings). Sets of Real Numbers. Topological Space. Metric Space. Convergence in R. Continuity in R. Convergence in Lp. Convergence in L2. Asymptotic Relations.
Generalized Functions
ALGEBRA
Elementary Algebra
Basic Algebra. Progressions. DeMoivre's Theorem. Partial Fractions.
Polynomials
Quadratic Polynomials. Cubic Polynomials. Quartic Polynomials. Quintic Polynomials. Tschirnhaus' Transformation. Polynomial Norms. Galois Group of a Polynomial. Other Polynomial Properties.
Number Theory
Congruences. Chinese Remainder Theorem. Continued Fractions. Diophantine Equations. Greatest Common Divisor. Least Common Multiple. Farey Sequences. Moebius Function. Prime Numbers. Prime Numbers less than 100,000. Prime Numbers of Special Forms. Prime Period Lengths. Factorization Table. Factorization of 2m-1. Magic Squares. Totient Function.
Vector Algebra
Notation for Vectors and Scalars. Physical Vectors. Fundamental Definitions. Laws of Vector Algebra. Vector Norms. Dot, Scalar, or Inner Product. Cross or Vector Product. Scalar and Vector Triple Products.
Linear and Matrix Algebra
Definitions. Types of Matrices. Conformability for Addition, Multiplication. Determinants. Matrix Norms. Singularity, Rank, and Inverses. Systems of Linear Equations. Other Matrix Transformations. Linear Spaces and Linear Mappings. Traces. Generalized Inverses. Eigenstructure. Eigenvalue Diagonalization. Matrix Exponentials. Quadratic Forms. Matrix Factorizations. Theorems. Kronecker Products or Tensor Products. Kronecker Sums. The Vector Operation.
Abstract Algebra
Basic Concepts. Groups. Rings. Fields. Finite Fields. Homomorphisms and Isomorphisms. Matrix Classes that are Groups. Permutation Groups. Primitive Normal Polynomials. Tables.
DISCRETE MATHEMATICS
Set Theory
Propositional Calculus. Tautologies. Truth Tables as Functions. Rules of Inference. Deductions. Sets. Set Operations and Relations. Venn Diagrams. Paradoxes and Theorems of Set Theory. Predicate Calculus.
Combinatorics
Sample Selection. Balls into Cells. Binomial Coefficients. Multinomial Coefficients. Arrangements and Derangements. Catalan Numbers. Partitions. Stirling Numbers. Stirling Cycle Numbers. Bell Numbers. Tables.
Graphs
Notation. Basic Definitions. Constructions. Fundamental Results. Tree Diagrams.
Partially Ordered Sets
Combinatorial Design Theory
t-Designs. Balanced Incomplete Block Diagrams (BIBDs). Difference Sets. Finite Geometry. Steiner Triple Systems. Hadamard Matrices. Latin Squares. Room Squares.
Inclusion/Exclusion
Communication Theory
Information Theory. Block Coding. Finite Fields. Binary Sequences. Morse Code. Source Coding for English Text.
Costas Array
Difference Equations
The Calculus of Finite Differences. Existence and Uniqueness. Linear Independence - General Solution. Homogenous Equations with Constant Coefficients. Nonhomogenous Equations with Constant Coefficients. Generating Functions and Z Transforms. Closed Form Solutions for Special Equations.
Discrete Dynamical Systems and Chaos
Chaotic One-Dimensional Maps. Logistic Map. Julia Sets and the Mandelbrot Set.
Operations Research
Linear Programming. Duality and Complementary Slackness. Linear Integer Programming. Branch and Bound. Network Flow Methods. Assignment Problem. Dynamic Programming. Shortest Path Problem.
GEOMETRY
Coordinate Systems in the Plane
Substitutions and Transformations. Cartesian Coordinates in the Plane. Polar Coordinates in the Plane. Homogenous Coordinates in the Plane. Oblique Coordinates in the Plane.
Symmetries or Isometries
Formulas for Symmetries: Cartesian Coordinates; Homogenous Coordinates; Polar Coordinates. Crystallographic Groups. Classifying the Crystallographic Groups.
Other Transformations of the Plane
Similarities. Affine Transformations. Projective Transformations.
Lines
Distances. Angles. Concurrence and Collinearity.
Polygons
Triangles. Quadrilaterals. Regular Polygons.
Circles
Conics
Alternative Characterization. The General Quadratic Equation. Additional Properties of Ellipses, Hyperbolas, and Parabolas.
Special Plane Curves
Algebraic Curves. Roulettes (Spirograph Curves). Spirals. The Peano Curve and Fractal Curves. Classical Constructions.
Coordinate Systems in Space
Cartesian Coordinates in Space. Cylindrical Coordinates in Space. Spherical Coordinates in Space. Relations between the Coordinates in Space. Homogenous Coordinates in Space.
Space Symmetries or Isometries
Formulae for Symmetries in Cartesian and Homogenous Coordinates.
Other Transformations of Space
Similarities. Affine Transformations. Projective Transformations.
Direction Angles and Direction Cosines
Planes
Lines
Polyhedra
Cylinders
Cones
Spheres
Surfaces of Revolution - The Torus
Quadrics
Knots up to Eight Crossings
Differential Geometry
Curves. Surfaces.
Angle Conversion
CONTINUOUS MATHEMATICS
Differential Calculus
Maxima and Minima of Functions. Vector Calculus. Matrix and Vector Derivatives.
Differential Forms
Integration
Definitions. Properties of Integration. Inequalities. Convergence Tests. Substitution. Partial Fraction Decomposition. Integration by Parts. Special Functions Defined by Integrals. Variational Principles. Line and Surface Integrals. Contour Integrals. Continuity of Integral Antiderivatives. Asymptotic Integral Evaluation. Moments of Inertia for Various Bodies. Table of Semi-Integrals. Tables of Integrals.
Table of Indefinite Integrals
Elementary Forms. Various Algebraic Forms. Forms Involving Trigonometric Functions. Forms Involving Inverse Trigonometric Functions. Logarithmic Forms. Exponential Forms. Hyperbolic Forms. Bessel Functions.
Table of Definite Integrals
Ordinary Differential Equations
Linear Differential Equations. Solution Techniques. Integrating Factors. Variation of Parameters. Green's Functions. List of Green's Functions. Transform Techniques. Named Ordinary Differential Equations. Liaponuv's Direct Method. Lie Groups. Types of Critical Points. Stochastic Differential Equations.
Partial Differential Equations
Classification of PDEs. Named Partial Differential Equations. Well-Posedness of PDEs. Green's Function. Quasi-Linear Equations. Exact Solutions to Laplace's Equation. Solutions to the Wave Equation. Separation of Variables. Transforming Partial Differential Equations.
Eigenvalues
Integral Equations
Definitions. Connection to Differential Equations. Fredholm Alternative. Sipecial Equations with Solutions.
Tensor Analysis
Definitions. Algebraic Tensor Operations. Differentiation of Tensors. Metric Tensors. Results. Examples.
Orthogonal Coordinate Systems
Control Theory
SPECIAL FUNCTIONS
Trigonometric or Circular Functions
Definition of Angles. Characterization of Angles. Relation between Radians and Degrees. Circular Functions. Other Trigonometric Functions. Periodicity Relations. Symmetry Relations. Signs in the Four Quadrants. Functions in Terms of the Angles in the First Quadrant. Circular Functions of Special Angles. Definitions in Terms of Exponentials. Fundamental Identities. Angle Sum and Difference Relations. Double Angle Formulae. Multiple Angle Formulae. Half Angle Formulae. Powers of Circular Functions. Products of Sine and Cosine. Sums of Circular Functions. Evaluating Sines and Cosines.
Circular Functions and Planar Triangles
Right Triangles. General Plane Triangles. Solution of Triangles.
Inverse Circular Functions
Definition. Fundamental Properties. Principle Values. Fundamental Identities. Functions of Negative Arguments. Relation to Inverse Hyperbolic Functions. Sum and Difference of Two Functions.
Spherical Geometry and Trigonometry
Right Spherical Trianges. Oblique Spherical Trianges. Table of Trigonometric Functions.
Exponential Functions
Exponentiation. Definition of ez. Derivative and Integral of ez. Circular Functions in Terms of Exponentials.
Logarithmic Functions
Definition of a Natural Log. Special Values. Logarithms to a Base Other than e. Relation of the Logarithm to the Exponential. Identities. Series Expansions for the Natural Logarithm. Derivative and Integration Formulae.
Hyperbolic Functions
Definitions. Range of Values. Series Expansions. Symmetry Relations. Interrelationships among the Hyperbolic Functions. Relation to Circular Functions. Hyperbolic Functions in Terms of One Another. Sum and Difference Formulae. Multiple Argument Relations. Sums and Products of Functions. Half-Argument Formulae. Differentiation Formulae.
Inverse Hyperbolic Functions
Range of Values. Relations among Inverse Hyperbolic Functions. Relations with Logarithmic and Circular Functions. Sum and Difference of Functions.
Gudermannian Function
Orthogonal Polynomials
Hermite Polynomials. Jacobi Polynomials. Laguerre Polynomials. Generalized Laguerre Polynomials. Legendre Polynomials. Tschebysheff Polynomials. Table of Jacobi Polynomials. Spherical Harmonics. Table of Spherical Harmonics.
The Gamma Function
Recursion Formula. Singular Points. Special Values. Definition by Products. Properties. Asymptotic Expansion.
The Beta Function
Error Functions and Fresnel Integrals
Series Expansions. Properties. Relationship with Normal Probability Function. Special Values. Asymptotic Expansion.
Sine, Cosine, and Exponential Integrals
Sine and Cosine Integrals. Alternative Definitions. Limits. Representations. Asymptotic Expansion. Exponential Integrals. Logarithmic Integral. Representations.
Polylogarithms
Definition. Singular Points. Integral. Generating Function. Special Values. Functional Equations for Dilogarithms.
Hypergeometric Functions
Pochhammer Symbol. Preparations. Definition. Properties. Polynomial Case. Special Cases. Special Values. Integral. Functional Relations. Differential Equation. Properties. Recursion Formulas.
Legendre Functions
Differential Equation. Definition. Polynomial Case. Singular Points. Relations. Special Case. Recursion Relations. Integrals. Relations between the Associated and Ordinary Legendre Functions. Orthogonality Relation.
Bessel Functions
Differential Equation. Singular Points. Relations. Series Expansions. Recurrence Relations. Behavior as z AE 0. Integrals. Fourier Expansion. Auxiliary Functions. Inverse Relations. Asymptotic Expansions. Zeros of Bessel Functions. Asymptotics of the Zeros. Half Order Bessel Functions. Modified Bessel Functions.
Elliptic Integrals
Definition. Elliptic Integral of the First, Second, and Third Kinds. Complete Elliptic Integral of the First and Second Kind. Complementary Integrals. Relations. Extensions of the Range of f Definition. Properties. Periods of the Elliptic Functions. Series Expansions.
Clebsch-Gordan Coefficients
Integral Transforms: Preliminaries
The Fourier Integral Transform
Existence. Properties. Inversion Formula. Poisson Summation Formula. Shannon's Sampling Theorem. Uncertainty Principle. Fourier Sine and Cosine Transforms.
Discrete Fourier Transform (DFT)
Fast Fourier Transform (FFT)
The Laplace Transform
Existence and Domain of Convergence. Inversion Formulas. Convolution.
The Z-Transform
Properties. Inversion Formula. Convolution and Product.
The Hilbert Transform
The Hankel Transform
Tables of Transforms
Multidimensional Fourier Transforms
PROBABILITY AND STATISTICS
Probability Theory
Introduction. Multivariate Distributions. Random Sums of Random Variables. Transforming Variables. Central Limit Theorem. Averages over Vectors. Inequalities. Geometric Probability. Classic Probability Problems.
Probability Distributions
Discrete Distributions. Continuous Distributions.
Queuing Theory
Markov Chains
Transition Function. Transition Matrix. Recurrence. Stationary Distributions. Random Walks. Ehrenfest Chain.
Random Number Generation
Methods of Pseudorandom Number Generation. Generating Nonuniform Random Variables.
Control Charts
Statistics
Descriptive Statistics. Statistical Estimators. Cramer-Rao Bound. Order Statistics. Classic Statistics Problems.
Confidence Intervals
Confidence Interval: Sample from One Population; Samples from Two Populations.
Test of Hypotheses
Hypothesis Tests: Parameter from One Population;
Parameters from Two Populations; Distribution of a Population; Distributions of Two Populations.
Sequential Probability Ratio Tests
Linear Regression
Linear Model y=b0 + b1x +x. General Model y=b0 + b1x1 + b2x2 +...+ bnxn +x.
Analysis of Variance (ANOVA)
One-Factor ANOVA. Unreplicated Two-Factor ANOVA. Replicated Two-Factor ANOVA.
Probability Tables
Critical Values. Table of the Normal Distribution. Percentage Points: t-Distribution; Chi-Square Distribution; F-Distribution. Cumulative Terms: Binomial Distribution; Poisson Distribution. Critical Values: Kolmogorov-Smirnov Test; Spearman's Rank Correlation.
Signal Processing
Estimation. Filters. Matched Filtering (Weiner Filter). Kalman Filtering. Walsh Functions. Wavelets.
SCIENTIFIC COMPUTING
Basic Numerical Analysis
Approximations and Errors. Solution to Algebraic Equations. Interpolation. Fitting Equations to Data. Numerical Linear Algebra. Gaussian Elimination. Nonlinear Systems and Numerical Optimization.
Numerical Integration and Differentiation
Numerical Integration. Numerical Differentiation. Schemes for the ODE: y' = f(x,y). Explicit formulas for the PDE: aux + ut = 0. Implicit formulas for the PDE: aux + ut = S(x,t). Schemes for the PDE: F(u)x + ut =0. Schemes for the PDE: ux=utt. Numerical Summation. Programming Techniques.
FINANCIAL ANALYSIS
Financial Formulae
Definition of Financial Terms. Formulae Connecting Financial Terms. Examples.
Financial Tables
Option Pricing
MISCELLANEOUS
Units
SI System of Measurement. Dimensional Analysis/Buckingham pi. Units of Physical Quantities. Conversion: Metric to English/English to Metric. Temperature Conversion. Miscellaneous Conversions. Physical Constants.
Calendar Computations
Leap Years. Day of the Week for Any Given Day. Number of Each Day of the Year.
AMS Classification Scheme
Greek Alphabet
Professional Mathematical Societies
Electronic Mathematical Resources
Computer Languages
Fields Medals
Biographies of Mathematicians
ASCII Character Codes
