72 articles
Extreme value theorem
The extreme value theorem, also called the Weierstrass extreme value theorem, states that a continuous function on a closed, bounded interval attains an absolute maximum and minimum.
Geometric series
A geometric series is an infinite series in which each term is a fixed multiple of the previous one, converging to a/(1−r) when the ratio's absolute value is below 1.
Hölder condition
A Hölder condition is a continuity requirement on a function, |f(x) − f(y)| ≤ A|x − y|^α with 0 < α ≤ 1; the case α = 1 is Lipschitz continuity.
Hypergeometric series
A hypergeometric series is a power series in which the ratio of consecutive coefficients is a rational function of the index, named by Euler in 1769 and made rigorous by Gauss in 1812.
Infimum and supremum
The infimum and supremum, also called the greatest lower bound and least upper bound, generalize the minimum and maximum of a set of real numbers.
Infinite product
An infinite product is a product of infinitely many factors, defined as the limit of its partial products, and is the multiplicative analog of an infinite series.
Intermediate value theorem
The intermediate value theorem, also known as Bolzano's theorem or the Cauchy–Bolzano theorem, states that a continuous real function on a closed interval takes every value between its endpoint values.
Interval (mathematics)
An interval in mathematics is a set of real numbers containing every number between any two of its members, such as the open and closed intervals.
John Edensor Littlewood
John Edensor Littlewood (1885–1977) was an English mathematician and Cambridge professor who worked on series, inequalities, and the zeta function, forming a famed partnership with G. H. Hardy.
Legendre transformation
The Legendre transformation, or Legendre transform, re-expresses a convex function in terms of its slope instead of its value, a key tool of convex analysis.
Limit (mathematics)
In mathematics, a limit is the value a function or sequence approaches as its input approaches some value, the foundational concept underlying continuity, derivatives, integrals, and convergence.
Limit of a function
The limit of a function is the value its output approaches as its input approaches a given point, and it underlies continuity, derivatives, and integrals in calculus.
Limit of a sequence
The limit of a sequence is the number its terms approach and stay near; a sequence with such a number is called convergent, one without, divergent.
Limit superior and limit inferior
The limit superior (limsup) and limit inferior (liminf) of a real sequence are its eventual upper and lower bounds, obtained from the suprema and infima of its tails.
Lipschitz continuity
Lipschitz continuity, the Lipschitz condition, is a strong form of uniform continuity in which a function cannot stretch distances by more than a fixed constant factor.
Modulus of continuity
The modulus of continuity of a function measures the largest amount it can change over pairs of points no farther apart than a given distance.
Monotone convergence theorem
The monotone convergence theorem, or Beppo Levi's theorem, is a result of Lebesgue integration proved in 1906 stating that the integral of the limit of increasing nonnegative functions equals the limit of their integrals.
Monotonic function
A monotonic function, also called a monotone function, is a real function that preserves or reverses order, so its outputs move in one fixed direction.
Natural logarithm
The natural logarithm, also called the natural log, is the logarithm to the base e, written ln x; it is the inverse of the exponential function.
Open ball
An open ball in a metric space is the set of all points within a given distance of a center, and it is always an open set.
p-series
A p-series is an infinite series of the form 1 + 1/2^p + 1/3^p + ... with real exponent p, converging exactly when p > 1 and otherwise diverging.
Peano curve
The Peano curve is a continuous curve that passes through every point of a square, constructed by Giuseppe Peano in 1890 as the first known space-filling curve.
Pointwise convergence
Pointwise convergence is a mode of convergence for sequences of functions where each point's values converge, without requiring the same rate everywhere; it is weaker than uniform convergence.
Power series
A power series is an infinite series of the form Σcₙ(x−a)ⁿ, an infinite polynomial that represents functions, converging within a radius of convergence R around its center.
Ratio test
The ratio test, also known as d'Alembert's ratio test, is a convergence test for infinite series, giving absolute convergence when the limit of consecutive-term ratios is below 1.
Real analysis
Real analysis, also called real variable analysis, is the rigorous branch of mathematics underlying calculus, built on the completeness of the real numbers and covering limits, continuity, differentiation, and integration.
Root test
The root test, also called Cauchy's root test or the nth-root test, decides whether an infinite series converges by examining the nth roots of its terms.
Semicontinuity
Semicontinuity is a one-sided weakening of continuity for real-valued functions, where upper or lower semicontinuity permits jumps in one direction; introduced by René Baire in 1899, it underlies existence of minimizers.
Sequence
A sequence, also called an ordered sequence, is a list of objects in a specified order, formally a function from the positive integers to the real numbers.
Series (mathematics)
A series, or infinite series, is the sum of the terms of a sequence, defined as the limit of its partial sums when that limit exists.