Flashcards: Mathematical analysis

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point.

In topology, the exterior of a subset S of a topological space X is the union of all open sets of X which are disjoint from S.

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved.

In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g); it produces a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change.

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

A finite difference is a mathematical expression of the form f(x + b) − f(x + a).

In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations (+ – × ÷), exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of nth roots).

In mathematics, division by zero is division where the divisor (denominator) is zero.

In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements of B, and A contains no greatest element.

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable).

In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form then for x in (x0, x1) the derivative of this integral is thus expressible as provided that f and its partial derivative fx are both continuous over a region in the form [x0, x1] × [y0, y1].

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others.

In mathematics, an analytic function is a function that is locally given by a convergent power series.

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S.

In mathematics, a real number is a value that represents a quantity along a line.

In mathematics, real coordinate space of n dimensions, written Rn (/ɑːrˈɛn/ ar-EN) (also written ℝn with blackboard bold) is a coordinate space that allows several (n) real variables to be treated as a single variable.

In mathematics, a metric space is a set for which distances between all members of the set are defined.

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.

In analytic geometry, an asymptote (/ˈæsɪmptoʊt/) of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity.

In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form where ƒ(x) is some twice-differentiable function, M is a large number, and the integral endpoints a and b could possibly be infinite.

In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure.

In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator.

In mathematics, a pullback is either of two different, but related processes: precomposition and fibre-product.

A Course of Modern Analysis (colloquially known as Whittaker and Watson) is a landmark textbook on mathematical analysis written by E.

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.

In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics.

In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0 or undefined.

In mathematics a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.

A polylogarithmic function in n is a polynomial in the logarithm of n, In computer science, polylogarithmic functions occur as the order of memory used by some algorithms (e.g., "it has polylogarithmic order").

A Course of Pure Mathematics is a classic textbook in introductory mathematical analysis, written by G.

In mathematics, a real-valued function or real function is a function whose values are real numbers.

In mathematical analysis, Darboux's formula is a formula introduced by Gaston Darboux () for summing infinite series by using integrals or evaluating integrals using infinite series.