Index

Definitions and theorems of Mathematical Analysis 1

The 79 definitions and 102 theorems of the theory of Mathematical Analysis 1, in the order of the topics. Each entry links to its box in the theory page.

Sets, induction and elementary functions

Definition Set, membership, inclusion
Definition Union, intersection, difference, complement
Definition Cartesian product
Definition The number sets N, Z, Q, R
Theorem Irrationality of the square root of 2
Definition Intervals and their notation
Definition Upper bound, lower bound, bounded set
Definition Maximum and minimum
Definition Supremum and infimum
Theorem Epsilon characterization of sup and inf
Theorem Completeness of R
Theorem Principle of induction
Theorem Sum of the first n natural numbers
Theorem Bernoulli's inequality
Definition Factorial
Definition Binomial coefficient
Theorem Properties of binomial coefficients
Theorem Binomial theorem
Definition Function, domain, codomain and range
Definition Injective, surjective, bijective function
Definition Inverse function
Definition Composite function
Definition Even function and odd function
Definition Periodic function
Definition Monotonic function
Definition Bounded function
Theorem Graph transformations
Definition Powers with natural and negative integer exponent
Definition Roots and powers with rational exponent
Definition Exponential function
Definition Logarithm and change of base
Definition Sine, cosine and tangent
Theorem Main identities
Definition Arcsine, arccosine and arctangent
Definition Hyperbolic functions
Definition Natural domain
Theorem Exponential and logarithmic equations and inequalities
Theorem Elementary trigonometric equations

Complex numbers

Definition Complex number
Definition Real part and imaginary part
Definition Sum, product, reciprocal and quotient
Definition Conjugate and its properties
Definition Modulus and its properties
Definition Complex plane (Gauss plane)
Definition Trigonometric form and argument
Theorem Product and quotient in trigonometric form
Theorem De Moivre's formula
Definition Euler's formula and exponential form
Theorem n-th roots of a complex number
Definition n-th roots of unity
Theorem Quadratic equations with real coefficients
Definition Equations with conjugate and modulus
Theorem Fundamental theorem of algebra
Theorem Complex conjugate roots of real polynomials
Theorem Real factorization of polynomials

Limits and continuity

Definition Neighborhood of a point
Definition Accumulation point
Definition Finite limit at a point
Definition Infinite limit and limit at infinity
Definition Right-hand limit and left-hand limit
Theorem Uniqueness of the limit
Theorem Sign-preservation theorem
Theorem Squeeze theorem
Theorem Bounded times infinitesimal
Theorem Limits of sums, products and quotients
Definition Indeterminate forms
Theorem Limit of a composite function
Theorem Fundamental sine limit
Theorem Fundamental exponential and logarithmic limits
Definition Infinitesimal, infinite, little-o
Definition Asymptotically equivalent functions
Theorem Hierarchy of infinities
Definition Asymptotes
Definition Continuous function
Theorem Continuity of elementary functions
Definition Points of discontinuity
Theorem Intermediate zero theorem
Theorem Weierstrass theorem
Theorem Intermediate value theorem

Differential calculus and Taylor

Definition Derivative and difference quotient
Definition One-sided derivatives
Definition Tangent line to the graph
Theorem Differentiable implies continuous
Definition Corner points, cusps, vertical-tangent inflection points
Theorem Derivative of sum, product and quotient
Theorem Derivative of a composite function
Theorem Derivative of the inverse function
Theorem Derivatives of the elementary functions
Definition Local maximum and minimum
Theorem Fermat's theorem
Theorem Rolle's theorem
Theorem Lagrange's theorem
Theorem Cauchy's theorem
Theorem Monotonicity and sign of the derivative
Theorem Zero derivative on an interval
Theorem Limit of the derivative
Theorem L'Hôpital's rule
Definition Higher-order derivatives and smoothness classes
Theorem Taylor's formula with Peano remainder
Theorem Taylor's formula with Lagrange remainder
Theorem Maclaurin expansions of the elementary functions
Definition Algebra of little-o

Graphing functions

Definition Scheme of curve sketching
Theorem Monotonicity test
Definition Stationary point
Definition Relative maximum and minimum
Theorem First derivative test
Theorem Second derivative test
Theorem Higher-order derivative test
Definition Absolute maximum and minimum
Definition Convex function and concave function
Theorem Convexity and second derivative
Theorem Position with respect to the tangent
Definition Inflection point
Theorem Necessary condition for an inflection point
Definition Sign chart of f' and f''
Definition Qualitative graph

Integral calculus

Definition Antiderivative and indefinite integral
Theorem Two antiderivatives differ by a constant
Theorem Elementary integrals
Theorem Elementary integrals in composite form
Theorem Integration by parts
Theorem Integration by substitution
Theorem Method of partial fractions
Definition Riemann definite integral
Theorem Classes of integrable functions
Theorem Properties of the definite integral
Theorem Mean value theorem for integrals
Definition Integral function
Theorem Fundamental theorem of calculus
Theorem Fundamental formula of calculus
Theorem Derivative of an integral function with variable limits
Theorem Area between two curves

Improper integrals

Definition Integral over an unbounded interval
Definition Integral of a function unbounded near an endpoint
Definition Integral with several critical points
Theorem Benchmark integrals of powers
Theorem Benchmark integral with the logarithm
Theorem Comparison test
Theorem Limit comparison test
Definition Order of infinitesimal and of infinity
Theorem Absolute convergence implies convergence
Theorem Three notable integrals

Numerical series

Definition Limit of a sequence
Theorem Bounded monotone sequences
Definition Series, convergence, sum
Theorem Geometric series
Definition Telescoping series
Theorem Harmonic series
Theorem Generalized harmonic series
Theorem Necessary condition for convergence
Theorem Behavior of series with nonnegative terms
Theorem Comparison test
Theorem Limit comparison test
Theorem Ratio test
Theorem Root test
Theorem Integral test
Theorem Absolute convergence
Theorem Leibniz test
Theorem Linearity of the sum
Definition Power series
Theorem Radius of convergence
Theorem Ratio and root for the radius
Theorem Elementary Taylor series

Differential equations

Definition Differential equation, order, solution
Definition General solution and particular solution
Definition Cauchy problem
Theorem Local existence and uniqueness
Definition Separable equation
Theorem Method of separation of variables
Definition First-order linear equation
Theorem Solution formula with the integrating factor
Theorem Structure of the general solution
Theorem Bernoulli equation
Definition Second-order equation with constant coefficients
Theorem Characteristic equation
Theorem Structure of the solution space
Theorem Method of undetermined coefficients
Theorem Variation of constants
Theorem Second-order Cauchy problem
Theorem Linear equations of order n with constant coefficients