List of real analysis topics


List of real analysis topics

This is a list of articles that are considered real analysis topics.

Contents

General topics

Limits

Sequences and Series

(see also list of mathematical series)

Summation Methods

More advanced topics

  • Convolution
  • 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

Convergence tests

Functions


Continuity

Distributions

Variation

Derivatives

Differentiation rules

Differentiation in Geometry and Topology

see also List of differential geometry topics

Integrals

(see also Lists of integrals)

  • 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

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

Specific Numbers

Sets

Maps

Applied mathematical tools

Infinite expressions

Inequalities

See list of inequalities

Means

Orthogonal polynomials

Spaces

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

Related fields of analysis


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