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Advanced Mathematics in Mathematica
3.1 Numbers
3.1.1 Types of Numbers
3.1.2 Numeric Quantities
3.1.3 Converting between Different Forms of Numbers
3.1.4 Numerical Precision
3.1.5 Arbitrary-Precision Numbers
3.1.6 Machine-Precision Numbers
3.1.7 Advanced Topic: Interval Arithmetic
3.1.8 Advanced Topic: Indeterminate and Infinite Results
3.1.9 Advanced Topic: Controlling Numerical Evaluation

3.2 Mathematical Functions
3.2.1 Naming Conventions
3.2.2 Numerical Functions
3.2.3 Pseudorandom Numbers
3.2.4 Integer and Number-Theoretical Functions
3.2.5 Combinatorial Functions
3.2.6 Elementary Transcendental Functions
3.2.7 Functions That Do Not Have Unique Values
3.2.8 Mathematical Constants
3.2.9 Orthogonal Polynomials
3.2.10 Special Functions
3.2.11 Elliptic Integrals and Elliptic Functions
3.2.12 Mathieu and Related Functions
3.2.13 Working with Special Functions
3.2.14 Statistical Distributions and Related Functions

3.3 Algebraic Manipulation
3.3.1 Structural Operations on Polynomials
3.3.2 Finding the Structure of a Polynomial
3.3.3 Structural Operations on Rational Expressions
3.3.4 Algebraic Operations on Polynomials
3.3.5 Polynomials Modulo Primes
3.3.6 Advanced Topic: Polynomials over Algebraic Number Fields
3.3.7 Trigonometric Expressions
3.3.8 Expressions Involving Complex Variables
3.3.9 Simplification

3.4 Manipulating Equations
3.4.1 The Representation of Equations and Solutions
3.4.2 Equations in One Variable
3.4.3 Advanced Topic: Algebraic Numbers
3.4.4 Simultaneous Equations
3.4.5 Equations Involving Functions
3.4.6 Getting Full Solutions
3.4.7 Advanced Topic: Existence of Solutions
3.4.8 Eliminating Variables
3.4.9 Solving Equations with Subsidiary Conditions
3.4.10 Advanced Topic: Solving Logical Combinations of Equations
3.4.11 Advanced Topic: Equations Modulo Integers

3.5 Calculus
3.5.1 Differentiation
3.5.2 Total Derivatives
3.5.3 Derivatives of Unknown Functions
3.5.4 Advanced Topic: The Representation of Derivatives
3.5.5 Defining Derivatives
3.5.6 Indefinite Integrals
3.5.7 Integrals That Can and Cannot Be Done
3.5.8 Definite Integrals
3.5.9 Manipulating Integrals in Symbolic Form
3.5.10 Differential Equations

3.6 Series, Limits and Residues
3.6.1 Making Power Series Expansions
3.6.2 Advanced Topic: The Representation of Power Series
3.6.3 Operations on Power Series
3.6.4 Advanced Topic: Composition and Inversion of Power Series
3.6.5 Converting Power Series to Normal Expressions
3.6.6 Solving Equations Involving Power Series
3.6.7 Summation of Series
3.6.8 Finding Limits
3.6.9 Residues

3.7 Linear Algebra
3.7.1 Constructing Matrices
3.7.2 Getting Pieces of Matrices
3.7.3 Scalars, Vectors and Matrices
3.7.4 Operations on Scalars, Vectors and Matrices
3.7.5 Multiplying Vectors and Matrices
3.7.6 Matrix Inversion
3.7.7 Basic Matrix Operations
3.7.8 Solving Linear Systems
3.7.9 Eigenvalues and Eigenvectors
3.7.10 Advanced Topic: Matrix Decompositions
3.7.11 Advanced Topic: Tensors

3.8 Numerical Operations on Data
3.8.1 Curve Fitting
3.8.2 Approximate Functions and Interpolation
3.8.3 Fourier Transforms

3.9 Numerical Operations on Functions
3.9.1 Numerical Mathematics in Mathematica
3.9.2 The Uncertainties of Numerical Mathematics
3.9.3 Numerical Integration
3.9.4 Numerical Evaluation of Sums and Products
3.9.5 Numerical Solution of Polynomial Equations
3.9.6 Numerical Root Finding
3.9.7 Numerical Solution of Differential Equations
3.9.8 Numerical Minimization
3.9.9 Linear Programming
3.9.10 Advanced Topic: Functions with Sensitive Dependence on Their Input

3.10 Mathematical and Other Notation
3.10.1 Special Characters
3.10.2 Names of Symbols and Mathematical Objects
3.10.3 Letters and Letter-like Forms
3.10.4 Operators
3.10.5 Structural Elements and Keyboard Characters