a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces (or more general spaces). Cauchy-continuous...

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as...

the idea of a Cauchy filter, in order to study completeness in topological spaces. The category of Cauchy spaces and Cauchy continuous maps is Cartesian...

derivatives of u and v satisfy the Cauchy–Riemann equations at that point. A holomorphic function is a complex function that is differentiable at every point...

mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies...

it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent...

mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat)...

holomorphic functions in D continuous up to the boundary of D. Then functions in H2(∂D) uniquely extend to holomorphic functions in D, and the Cauchy integral...

Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function ⁠ f {\displaystyle f} ⁠, this is equivalent to...

condition Cauchy bounds Cauchy completeness Cauchy completion Cauchy condensation test Cauchy-continuous function Cauchy's convergence test Cauchy (crater)...

In mathematics, the Cauchy principal value, named after Augustin-Louis Cauchy, is a method for assigning values to certain improper integrals which would...

if f {\displaystyle f} is Cauchy-continuous. It is easy to see that every uniformly continuous function is Cauchy-continuous and thus extends to X {\displaystyle...

arguments were introduced into calculus. Here Cauchy defined continuity as follows: The function f(x) is continuous with respect to x between the given limits...

mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain...

The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the inner product between two vectors in an inner...

analysis, continuous wavelets are functions used by the continuous wavelet transform. These functions are defined as analytical expressions, as functions either...

which is related to the cumulative distribution function of a Cauchy distribution. A sigmoid function is constrained by a pair of horizontal asymptotes...

a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given...

result from the previous section. This is the characteristic function of the standard Cauchy distribution: thus, the sample mean has the same distribution...

signal. Continuous wavelet S transform Time-frequency analysis Cauchy wavelet A. Grossmann & J. Morlet, 1984, Decomposition of Hardy functions into square...

mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively...

characteristic function of a continuous random variable X {\displaystyle X} is the Fourier transform of its probability density function f X ( x ) {\displaystyle...

known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions...

the volume of the body is assumed to be continuous. Therefore, there exists a contact force density or Cauchy traction field T ( n , x , t ) {\displaystyle...

In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given...

approximations are cumulative distribution functions of common probability distributions: the logistic, Cauchy and normal distributions, respectively. Approximations...

Instead of working with Cauchy sequences, one works with Cauchy filters (or Cauchy nets). A Cauchy filter (respectively, a Cauchy prefilter) F {\displaystyle...

an additive function f : R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } is linear if: f {\displaystyle f} is continuous (Cauchy, 1821). In...

) or inverse distribution function. With reference to a continuous and strictly monotonic cumulative distribution function (c.d.f.) F X : R → [ 0 , 1...

define continuous functions. However, his work remained unknown to other mathematicians until thirty years after his death. Augustin-Louis Cauchy in 1821...