(physics) A function, denoted by Ai(x), that is a solution to the Airy equation.

(figuratively) A system or process, that is like algebra by substituting one thing for another, or in using signs, symbols, etc., to represent concepts or ideas.

(mathematics) A function obtained from a continuous function by transfinite iteration of the operation of forming pointwise limits of sequences of functions.

(mathematics) A method for reconstructing a harmonic function.

(mathematics) Any of a class of functions that are solutions to a particular form of differential equation (a Bessel equation) and are typically used to describe waves in a cylindrically symmetric system.

(mathematics) A set whose boundary is measurable and has (at least locally) finite measure.

(mathematics, complex analysis, always plural) Given a complex-valued function f and real-valued functions u and v such that f(z) = u(z) + iv(z), either of the equations (∂u)/(∂x)=(∂v)/(∂y) or (∂u)/(∂y)=-(∂v)/(∂x), which together form part of the criteria that f be complex-differentiable.

(mathematical analysis) Of two mappings, a point in the domain of both mappings that has same image under both.

(mathematics) A method of finding an approximate solution of an ordinary differential equation L[y]=0 by determining coefficients in an expansion y(x)=y_0(x)+∑ₗ₌₀^q𝛼ₗy_l(x) so as to make L[y] vanish at prescribed points; the expansion with the coefficients thus found is the sought approximation.

(mathematics, complex analysis, of a function) That is differentiable and satisfies the Cauchy-Riemann equations on a subset of the complex plane.

(mathematical analysis) a function whose value at any point in its domain is equal its limit at the same point

(algebra) The variety defined by a covariant.

(generally) The linear operator that maps functions to their derived functions, usually written D; the simplest differential operator.

(mathematics, informal) Synonym of Cantor function; a type of monotonic increasing function.

(number theory) A function, denoted by ρ, used to estimate the proportion of smooth numbers up to a given bound.

(mathematics) A function of one real argument, whose value is zero when the argument is nonzero, and whose integral is one over any interval that includes zero.

(mathematical analysis, functional analysis, Fourier analysis) A quadratic functional which, given a real function defined on an open subset of ℝⁿ, yields a real number that is a measure of how variable said function is.

(mathematics) A function used to predict a categorical dependent variable by one or more continuous or binary independent variables.

(mathematics) The set of all possible mathematical entities (points) where a given function is defined.

(mathematics, numerical analysis) A method for solving ordinary differential equations, using six function evaluations to calculate fourth- and fifth-order accurate solutions, the difference between which is then taken to be the error of the (fourth-order) solution.

(mathematics) An exact, nonlinear solution of any of a large class of two-dimensional partial differential equations

(mathematics) Any function of a complex variable that is holomorphic throughout the complex plane

(mathematics) The real number system adjoined with two extra symbols: ∞ and −∞, inheriting the ordering of the real number system, and defining −∞ to be less than any real number, and ∞ to be larger than any real number.

(mathematics) Either of two mathematical constants that express ratios in a bifurcation diagram for a nonlinear map.

(mathematics) The preimage of a given point in the range of a map.

(mathematics, order theory) A non-empty upper set (of a partially ordered set) which is closed under binary infima (a.k.a. meets).

(mathematics, set theory, probability theory) A totally ordered collection of subsets.

(mathematical analysis, especially nonstandard analysis) An ultrafilter which contains all cofinite subsets of the set which is being "filtered" by it.

(mathematics) A certain kind of subset of a partially ordered set.

(calculus) A generalization of the fundamental theorem of calculus to the two-dimensional plane, which states that given two scalar fields P and Q and a simply connected region R, the area integral of derivatives of the fields equals the line integral of the fields, or

(mathematics) A rational number that, in certain situations, counts pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. They have applications in string theory.

(mathematics) A particular function, gd(x)=2 tan ⁻¹eˣ-𝜋/2.

(mathematics) A type of measure of central tendency calculated as the reciprocal of the mean of the reciprocals, ie, H=n/1/x_1+1/x_2+⋯+1/x_n

(mathematics) A theorem stating that a sequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergent subsequence.

(mathematics) Having the property that if a sequence of points in the domain of a function converges to a point L, then either the sequence of sets that are the images of those points contains a sequence that converges to a point that is in the image of L, or, alternatively, for every element in the image of L, you can find a sub-sequence in the domain whose image contains a convergent sequence to that element.

(mathematics) The tendency of high-dimensional data to contain points (hubs) that frequently occur in k-nearest-neighbor lists of other points.

(mathematics, of a function) the set of all points lying on or below its graph.

(algebra, computing) Any function which maps all elements of its domain to themselves.

(mathematics) A function whose value is always the same as its independent variable, and for which the codomain equals the domain.

(mathematics) What a function maps to.

(mathematics) A function whose domain is a subset of its codomain, and for which all of the elements in its domain are fixed points.

(mathematics) A function between two sets A and B, with the former being a subset of the latter, that maps elements from A to the corresponding elements of B (𝜄:A→B,𝜄(x)=x).

(mathematical analysis, of a Borel set) Whose measure is equal to the supremum of the measures of all compact sets which are contained by it.

(mathematics) An inequality that relates the value of a convex function of an integral to the integral of the convex function.

(mathematics) The second-order linear differential equation xy+(1-x)y'+ny=0.

(mathematics, physics) A differential operator,denoted ∆ and defined on ℝⁿ as 𝛥=∑ᵢ₌₁ⁿ(∂²)/(∂x_i²), used in the modeling of wave propagation, heat flow and many other applications.

(mathematics) an integral transform of positive real function f(t) to a complex function F(s); given by:

(mathematics) A certain operation on a measure space; see lifting theory.

(mathematics) An integral whose integrand's domain is a curve.

(mathematical analysis) A property which can be said to be held by some point in the domain of a real-valued function if there exists a neighborhood of that point and a certain constant such that for any other point in that neighborhood, the absolute value of the difference of their function values is less than the product of the constant and the absolute value of the difference between the two points.

(calculus) Any Taylor series that is centred at 0 (i.e., for which the origin is the reference point used to derive the series from its associated function); for a given infinitely differentiable complex function f, the power series f(0)+(f'(0))/(1!)x+(f(0))/(2!)x²+(f'(0))/(3!)x³+⋯=∑ₙ₌₀ ᪲(f⁽ⁿ⁾(0))/(n!),xⁿ.

(mathematics) A function that maps every element of a given mathematical structure (eg: a set) to a unique element of another structure; a correspondence.

(mathematical analysis, number theory, statistics) An integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform.

(complex analysis) A complex function defined as E_𝛼,𝛽(z)=∑ₖ₌₀ ᪲(zᵏ)/(𝛤(𝛼k+𝛽)), where 𝛤(x) is the gamma function.

(complex analysis) A part of the theory of meromorphic functions that describes the asymptotic distribution of solutions to the equation ƒ(z) = a, as a varies.

(calculus) An equation involving the derivatives of a function of only one independent variable.

(mathematical analysis) (of a Borel set) That its measure is equal to the infimum of the measures of all open sets which contain it.

(mathematics) A formula for calculating the sum of an arithmetical function by means of an inverse Mellin transform.

(mathematical analysis) A function whose domain is a σ-algebra, whose codomain is [0,∞], and which is countably additive.

(mathematics, mathematical analysis) Any infinite series of the general form ∑ᵢ₌₀ ᪲a_i(x-c)ⁱ.

(mathematics) For a given function, the set of all elements of the domain that are mapped into a given subset of the codomain; (formally) given a function ƒ : X → Y and a subset B ⊆ Y, the set ƒ⁻¹(B) = {x ∈ X : ƒ(x) ∈ B}.

𝜌(g,h)(0,x_1,…,x_k)=g(x_1,…,x_k)\𝜌(g,h)(y+1,x_1,…,x_k)=h(y,𝜌(g,h)(y,x_1,…,x_k),x_1,…,x_k),.

(mathematics) A measure on the σ-algebra of Borel sets of a Hausdorff space that is locally finite and inner regular.

(mathematics) a meromorphic function

(mathematics) A statement of equality of two products of generators, used in the presentation of a group.

(complex analysis) A form of complex number, proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities.

(mathematics) A formal system in which index notation is used to define tensors and tensor fields and the rules for their manipulation; the theory of tensor calculus as developed by Gregorio Ricci-Curbastro, which formed the foundation of the modern theory.

(mathematics) A certain kind of approximation of an integral by a finite sum that is calculated by dividing the interval of integration into smaller sub-intervals and summing sample values of the integrand inside those sub-intervals multiplied by their lengths.

(number theory, analytic number theory, uncountable) The function ζ defined by the Dirichlet series 𝜁(s)=∑ₙ₌₁ ᪲1/(nˢ)=1/(1ˢ)+1/(2ˢ)+1/(3ˢ)+1/(4ˢ)+⋯, which is summable for points s in the complex half-plane with real part > 1; the analytic continuation of said function, being a holomorphic function defined on the complex numbers with pole at 1.

Alternative spelling of Riemann zeta function [(number theory, analytic number theory, uncountable) The function ζ defined by the Dirichlet series 𝜁(s)=∑ₙ₌₁ ᪲1/(nˢ)=1/(1ˢ)+1/(2ˢ)+1/(3ˢ)+1/(4ˢ)+⋯, which is summable for points s in the complex half-plane with real part > 1; the analytic continuation of said function, being a holomorphic function defined on the complex numbers with pole at 1.]

(calculus) The theorem that any real-valued differentiable function that attains equal values at two distinct points must have a point somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero. In mathematical terms, if f:ℝ→ℝ is differentiable on (a,b) and f(a)=f(b) then ∃c∈(a,b):f'(c)=0.

(mathematics) Any function whose domain is a vector space and whose value is its scalar field

(mathematics) A branch of algebraic geometry concerned with solving certain types of counting problem in projective geometry; a symbolic calculus used to represent and solve such problems;

(mathematics) A subscript

(mathematics) A type of function such that for every exponent a, there is a small integer ε such that for any positive value x less than ε, its image lies between xᵃ and x⁻ᵃ .

(mathematics)

(mathematics) The domain of a signature morphism (a mapping of a set of symbols of a given signature into another set of symbols) that is an inclusion.

(mathematics) The sum of a subset of values.

(mathematics) The set of all sums of an element from A with an element from B, where A and B are subsets of an abelian group.

(mathematics) A superalgebra over a ring.

(calculus) A power series representation of given infinitely differentiable function f whose terms are calculated from the function's arbitrary order derivatives at given reference point a; the series f(a)+(f'(a))/(1!)(x-a)+(f(a))/(2!)(x-a)²+(f'(a))/(3!)(x-a)³+⋯=∑ₙ₌₀∞(f⁽ⁿ⁾(a))/(n!)(x-a)ⁿ.

(mathematics) Totally ordered multiset.

(specifically, complex analysis) A meromorphic function on the complex plane that is not a rational function.

(mathematics)

(mathematics) A continuous linear functional.

(mathematics) A type of generalized non-Archimedean function

(mathematical analysis) The metric 𝜌(f,g)= sup _(x∈X) where f and g are functions on X.

(mathematical analysis, of a function from a metric space X to a metric space Y) That for every real ε > 0 there exists a real δ > 0 such that for all pairs of points x and y in X for which D_X(x,y)<𝛿, it must be the case that D_Y(f(x),f(y))<𝜀 (where D_X and D_Y are the metrics of X and Y, respectively).

(measure theory, domain theory) A map from the class of open sets of a topological space to the set of positive real numbers including infinity.

(mathematics) Any member of a (generalized) vector space.

(mathematics) Any function whose range is n-dimensional

(mathematics) function of the complex variable s that analytically continues the sum of the infinite series ∑ₙ₌₁ ᪲1/(nˢ) that converges when the real part of s is greater than 1.

(mathematical analysis) A σ-algebra which is obtained as a "completion" of a given σ-algebra, which includes all subsets of the given measure space which simultaneously contain a member of the given σ-algebra and are contained by a member of the given σ-algebra, as long as the contained and containing measurable sets have the same measure, in which case the subset in question is assigned a measure equal to the common measure of its contained and containing measurable sets (so the measure is also being completed, in parallel with the σ-algebra).