• a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces (or more general spaces). Cauchy-continuous...
    5 KB (826 words) - 19:55, 11 September 2023
  • Thumbnail for Cauchy distribution
    The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as...
    46 KB (6,910 words) - 18:35, 11 July 2025
  • 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...
    63 KB (9,324 words) - 15:49, 8 July 2025
  • Thumbnail for Uniform continuity
    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...
    25 KB (4,170 words) - 00:42, 30 June 2025
  • 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...
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  • Thumbnail for Dirac delta function
    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...
    98 KB (14,418 words) - 05:48, 22 July 2025
  • Thumbnail for Cauchy–Riemann equations
    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...
    34 KB (5,011 words) - 18:33, 3 July 2025
  • Thumbnail for Augustin-Louis Cauchy
    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...
    42 KB (5,401 words) - 03:26, 30 June 2025
  • Thumbnail for Holomorphic function
    Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function ⁠ f {\displaystyle f} ⁠, this is equivalent to...
    25 KB (3,490 words) - 21:26, 15 June 2025
  • In mathematics, the Cauchy principal value, named after Augustin-Louis Cauchy, is a method for assigning values to certain improper integrals which would...
    11 KB (1,966 words) - 02:32, 14 June 2025
  • 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...
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  • Thumbnail for Cauchy's integral theorem
    mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat)...
    10 KB (1,643 words) - 15:23, 27 May 2025
  • Thumbnail for Cauchy's integral formula
    it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent...
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  • 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...
    12 KB (2,570 words) - 23:28, 24 July 2025
  • Thumbnail for Heaviside step function
    approximations are cumulative distribution functions of common probability distributions: the logistic, Cauchy and normal distributions, respectively. Approximations...
    14 KB (2,157 words) - 11:06, 13 June 2025
  • Let f : [ a , b ] → R {\displaystyle f:[a,b]\to \mathbb {R} } be a continuous function on the closed interval [ a , b ] {\displaystyle [a,b]} , and differentiable...
    28 KB (5,401 words) - 07:03, 18 July 2025
  • Thumbnail for Characteristic function (probability theory)
    result from the previous section. This is the characteristic function of the standard Cauchy distribution: thus, the sample mean has the same distribution...
    38 KB (5,208 words) - 13:53, 16 April 2025
  • mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain...
    6 KB (963 words) - 11:19, 28 June 2025
  • The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between...
    37 KB (5,182 words) - 23:37, 5 July 2025
  • functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X {\displaystyle X} with values in the...
    7 KB (1,110 words) - 08:18, 17 April 2025
  • Augustin-Louis Cauchy include: Bolzano–Cauchy theorem Cauchy boundary condition Cauchy completion Cauchy-continuous function Cauchy–Davenport theorem Cauchy distribution...
    3 KB (205 words) - 10:51, 15 May 2025
  • analysis, continuous wavelets are functions used by the continuous wavelet transform. These functions are defined as analytical expressions, as functions either...
    2 KB (169 words) - 03:37, 12 November 2024
  • Thumbnail for Sigmoid function
    which is related to the cumulative distribution function of a Cauchy distribution. A sigmoid function is constrained by a pair of horizontal asymptotes...
    16 KB (2,095 words) - 12:59, 12 July 2025
  • Thumbnail for Fourier transform
    {\displaystyle \xi } produces the frequency-domain function, and it converges at all frequencies to a continuous function tending to zero at infinity. If f ( x )...
    177 KB (21,313 words) - 19:14, 8 July 2025
  • Thumbnail for Probability density function
    a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given...
    30 KB (4,947 words) - 17:28, 9 July 2025
  • locally the graph of a function. Augustin-Louis Cauchy (1789–1857) is credited with the first rigorous form of the implicit function theorem. Ulisse Dini...
    23 KB (3,821 words) - 05:35, 7 June 2025
  • known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions...
    19 KB (2,675 words) - 22:24, 4 June 2025
  • Thumbnail for Quantile function
    distribution function or c.d.f.) or inverse distribution function. With reference to a continuous and strictly increasing cumulative distribution function (c.d...
    17 KB (2,217 words) - 11:33, 12 July 2025
  • Thumbnail for Karl Weierstrass
    1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement that is...
    17 KB (1,662 words) - 22:36, 19 June 2025
  • characteristic function of a continuous random variable X {\displaystyle X} is the Fourier transform of its probability density function f X ( x ) {\displaystyle...
    19 KB (2,820 words) - 18:06, 19 July 2025