- List of real analysis topics
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This is a list of articles that are considered real analysis topics.
Contents
General topics
Limits
- Limit of a sequence
- Subsequential limit - the limit of some subsequence
- Limit of a function (see List of limits for a list of limits of common functions)
- One-sided limit - either of the two limits of functions of real variables x, as x approaches a point from above or below
- Squeeze theorem - confirms the limit of a function via comparison with two other functions
- Big O notation - used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions
Sequences and Series
(see also list of mathematical series)
- Arithmetic progression - a sequence of numbers such that the difference between the consecutive terms is constant
- Generalized arithmetic progression - a sequence of numbers such that the difference between consecutive terms can be one of several possible constants
- Geometric progression - a sequence of numbers such that each consecutive term is found by multiplying the previous one by a fixed non-zero number
- Harmonic progression - a sequence formed by taking the reciprocals of the terms of an arithmetic progression
- Finite sequence - see sequence
- Infinite sequence - see sequence
- Divergent sequence - see limit of a sequence or divergent series
- Convergent sequence - see limit of a sequence or convergent series
- Cauchy sequence - a sequence whose elements become arbitrarily close to each other as the sequence progresses
- Convergent series - a series whose sequence of partial sums converges
- Divergent series - a series whose sequence of partial sums diverges
- Power series - a series of the form
- Taylor series - a series of the form
- Maclaurin series - see Taylor series
- Binomial series - the Maclaurin series of the function f given by f(x) = (1 + x) α
- Maclaurin series - see Taylor series
- Taylor series - a series of the form
- Telescoping series
- Alternating series
- Geometric series
- Harmonic series
- Fourier series
- Lambert series
Summation Methods
- Cesàro summation
- Euler summation
- Lambert summation
- Borel summation
- Summation by parts - transforms the summation of products of into other summations
- Cesàro mean
- Abel's summation formula
More advanced topics
- Convolution
- Cauchy product - is the discrete convolution of two sequences
- Farey sequence - the sequence of completely reduced fractions between 0 and 1
- Oscillation - is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.
- Indeterminate forms - algerbraic expressions gained in the context of limits. The indeterminate forms include 00, 0/0, 1∞, ∞ − ∞, ∞/∞, 0 × ∞, and ∞0.
Convergence
- Pointwise convergence, Uniform convergence
- Absolute convergence, Conditional convergence
- Normal convergence
Convergence tests
- Integral test for convergence
- Cauchy's convergence test
- Ratio test
- Comparison test
- Root test
- Alternating series test
- Cauchy condensation test
- Abel's test
- Dirichlet's test
- Stolz–Cesàro theorem - is a criterion for proving the convergence of a sequence
Functions
- Function of a real variable
- Continuous function
- Smooth function
- Differentiable function
- Integrable function
- Square-integrable function, p-integrable function
- Monotonic function
- Bernstein's theorem on monotone functions - states that any real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions
- Inverse function
- Convex function, Concave function
- Singular function
- Harmonic function
- Rational function
- Orthogonal function
- Implicit and explicit functions
- Implicit function theorem - allows relations to be converted to functions
- Measurable function
- Baire one star function
- Symmetric function
Continuity
- Uniform continuity
- Semi-continuity
- Equicontinuous
- Absolute continuity
- Hölder condition - condition for Hölder continuity
Distributions
Variation
Derivatives
- Second derivative
- Inflection point - found using second derivatives
- Directional derivative, Total derivative, Partial derivative
Differentiation rules
- Linearity of differentiation
- Product rule
- Quotient rule
- Chain rule
- Inverse function theorem - gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain, also gives a formula for the derivative of the inverse function
Differentiation in Geometry and Topology
see also List of differential geometry topics
- Differentiable manifold
- Differentiable structure
- Submersion - a differentiable map between differentiable manifolds whose differential is everywhere surjective
Integrals
(see also Lists of integrals)
- Antiderivative
- Fundamental Theorem of Calculus - a theorem of anitderivatives
- Multiple integral
- Iterated integral
- Improper integral
- Cauchy principal value - method for assigning values to certain improper integrals
- Line integral
- Anderson's theorem - says that the integral of an integrable, symmetric, unimodal, non-negative function over an n-dimensional convex body (K) does not decrease if K is translated inwards towards the origin
Integration and Measure theory
see also List of integration and measure theory topics
- Riemann integral , Riemann sum
- Riemann–Stieltjes integral
- Darboux integral
- Lebesgue integration
Fundamental theorems
- Monotone convergence theorem - relates monotonicity with convergence
- Intermediate value theorem - states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value
- Rolle's theorem - essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero
- Mean value theorem - that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc
- Taylor's theorem - gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial.
- L'Hopital's rule - uses derivatives to help evaluate limits involving indeterminate forms
- Abel's theorem - relates the limit of a power series to the sum of its coefficients
- Lagrange inversion theorem - gives the taylor series of the inverse of an analytic function
- Darboux's theorem - states that all functions that result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval
- Heine-Borel theorem - sometimes used as the defining property of compactness
- Bolzano-Weierstrass theorem - states that each bounded sequence in Rn has a convergent subsequence.
Foundational topics
Numbers
Real numbers
- Construction of the real numbers
- Completeness of the real numbers
- Least-upper-bound property
- Real line
Specific Numbers
Sets
Maps
- Contraction mapping
- Metric map
- Fixed point - a point of a function that maps to itself
Applied mathematical tools
Infinite expressions
Inequalities
- Triangle inequality
- Bernoulli's inequality
- Cauchy-Schwarz inequality
- Triangle inequality
- Hölder's inequality
- Minkowski inequality
- Jensen's inequality
- Chebyshev's inequality
- Inequality of arithmetic and geometric means
Means
- Generalized mean
- Pythagorean means
- Geometric-harmonic mean
- Arithmetic-geometric mean
- Weighted mean
- Quasi-arithmetic mean
Orthogonal polynomials
Spaces
- Euclidean space
- Metric space
- Banach fixed point theorem - guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, provides method to find them
- Complete metric space
- Topological space
- Compact space
Measures
- Dominated convergence theorem - provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions.
Field of sets
Historical figures
- Michel Rolle (1652-1719)
- Brook Taylor (1685-1731)
- Leonhard Euler (1707-1783)
- Joseph Louis Lagrange (1736-1813)
- Jean Baptiste Joseph Fourier (1768-1830)
- Bernard Bolzano (1781-1848)
- Augustin Cauchy (1789-1857)
- Niels Henrik Abel (1802-1829)
- Johann Peter Gustav Lejeune Dirichlet (1805-1859)
- Karl Weierstrass (1815-1897)
- Eduard Heine (1821-1881)
- Pafnuty Chebyshev (1821-1894)
- Leopold Kronecker (1823-1891)
- Bernhard Riemann (1826-1866)
- Richard Dedekind (1831-1916)
- Rudolf Lipschitz (1832-1903)
- Camille Jordan (1838-1922)
- Jean Gaston Darboux (1842-1917)
- Georg Cantor (1845-1918)
- Ernesto Cesàro (1859-1906)
- Otto Hölder (1859-1937)
- Hermann Minkowski (1864-1909)
- Alfred Tauber (1866-1942)
- Felix Hausdorff (1868-1942)
- Émile Borel (1871-1956)
- Henri Lebesgue (1875-1941)
- Waclaw Sierpinski (1882-1969)
- Johann Radon (1887-1956)
- Karl Menger (1902-1985)
Related fields of analysis
- Asymptotic analysis - studies a method of describing limiting behaviour
- Convex analysis - studies the properties of convex functions and convex sets
- Harmonic analysis - studies the representation of functions or signals as superpositions of basic waves
- Fourier analysis - studies Fourier series and Fourier transforms
- Complex analysis - studies the extension of real analysis to include complex numbers
- Functional analysis - studies vector spaces endowed with limit-related structures and the linear operators acting upon these spaces
Categories:- Real analysis
- Outlines
- Mathematics-related lists
- Limit of a sequence
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