* 1789 – Augustin-Louis Cauchy, French mathematician ( d. 1857 )

A firmly non-expansive mapping is always non-expansive, via the Cauchy – Schwarz inequality.

It is also known, especially among physicists, as the Lorentz distribution ( after Hendrik Lorentz ), Cauchy – Lorentz distribution, Lorentz ( ian ) function, or Breit – Wigner distribution.

In mathematics, the Cauchy – Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable.

The Cauchy – Riemann equations on a pair of real-valued functions of two real variables u ( x, y ) and v ( x, y ) are the two equations:

Then f = u + iv is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy – Riemann equations ( 1a ) and ( 1b ) at that point.

The sole existence of partial derivatives satisfying the Cauchy – Riemann equations is not enough to ensure complex differentiability at that point.

The reason why Euler and some other authors relate the Cauchy – Riemann equations with analyticity is that a major theorem in complex analysis says that holomorphic functions are analytic and viceversa.

First, the Cauchy – Riemann equations may be written in complex form

Consequently, a function satisfying the Cauchy – Riemann equations, with a nonzero derivative, preserves the angle between curves in the plane.

That is, the Cauchy – Riemann equations are the conditions for a function to be conformal.

which are the Cauchy – Riemann equations ( 2 ) at the point z < sub > 0 </ sub >.

Conversely, if ƒ: C → C is a function which is differentiable when regarded as a function on R < sup > 2 </ sup >, then ƒ is complex differentiable if and only if the Cauchy – Riemann equations hold.

In other words, if u and v are real-differentiable functions of two real variables, obviously u + iv is a ( complex-valued ) real-differentiable function, but u + iv is complex-differentiable if and only if the Cauchy – Riemann equations hold.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

But this is exactly the Cauchy – Riemann equations, thus ƒ is differentiable at 0 if and only if the Cauchy – Riemann equations hold at 0.

The above proof suggests another interpretation of the Cauchy – Riemann equations.

The Cauchy – Riemann equations can then be written as a single equation

In this form, the Cauchy – Riemann equations can be interpreted as the statement that f is independent of the variable.

In mathematics, the Cauchy – Schwarz inequality ( also known as the Bunyakovsky inequality, the Schwarz inequality, or the Cauchy – Bunyakovsky – Schwarz inequality, or Cauchy – Bunyakovsky inequality ), is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas.

and after multiplication by || v ||< sup > 2 </ sup > the Cauchy – Schwarz inequality.

In Euclidean space R < sup > n </ sup > with the standard inner product, the Cauchy – Schwarz inequality is

which yields the Cauchy – Schwarz inequality.

When n = 3 the Cauchy – Schwarz inequality can also be deduced from Lagrange's identity, which takes the form

from which readily follows the Cauchy – Schwarz inequality.

The triangle inequality for the inner product is often shown as a consequence of the Cauchy – Schwarz inequality, as follows: given vectors x and y:

The Cauchy – Schwarz inequality allows one to extend the notion of " angle between two vectors " to any real inner product space, by defining:

The Cauchy – Schwarz inequality proves that this definition is sensible, by showing that the right hand side lies in the interval, and justifies the notion that ( real ) Hilbert spaces are simply generalizations of the Euclidean space.

The Cauchy – Schwarz is used to prove that the inner product is a continuous function with respect to the topology induced by the inner product itself.

The Cauchy – Schwarz inequality is usually used to show Bessel's inequality.

and so, by the Cauchy – Schwarz inequality,

Various generalizations of the Cauchy – Schwarz inequality exist in the context of operator theory, e. g. for operator-convex functions, and operator algebras, where the domain and / or range of φ are replaced by a C *- algebra or W *- algebra.

In this language, the Cauchy – Schwarz inequality becomes

We now give an operator theoretic proof for the Cauchy – Schwarz inequality which passes to the C *- algebra setting.

The finite-dimensional case of this inequality for real vectors was proved by Cauchy in 1821, and in 1859 Cauchy's student Bunyakovsky noted that by taking limits one can obtain an integral form of Cauchy's inequality.

Mathematically, the conjecture states that, for generic initial data, the maximal Cauchy development possesses a complete future null infinity.

Mathematically, the conjecture states that the maximal Cauchy development of generic compact or asymptotically flat initial data is locally inextendible as a regular Lorentzian manifold.

The Cauchy – Kowalevski theorem states that the Cauchy problem for any partial differential equation whose coefficients are analytic in the unknown function and its derivatives, has a locally unique analytic solution.

This principle is critical in determinism, which in the language of general relativity states complete knowledge of the universe on a spacelike Cauchy surface can be used to calculate the complete state of the rest of spacetime.

The geodesic minimizes the entropy, due to the Cauchy – Schwarz inequality, which states that the action is bounded below by the length of the curve, squared.

The utility of Cauchy sequences lies in the fact that in a complete metric space ( one where all such sequences are known to converge to a limit ), the criterion for convergence depends only on the terms of the sequence itself.

of real numbers is called Cauchy, if for every positive real number ε, there is a positive integer N such that for all natural numbers m, n > N

The derivative of the quantile function, the quantile density function, for the Cauchy distribution is:

and this is its Cauchy principal value, which is zero, but we could also take ( 1 ) to mean, for example,

The log-likelihood function for the Cauchy distribution for sample size n is:

and expresses the two parameters of the associated linear Cauchy distribution for x as a complex number:

See also McCullagh's parametrization of the Cauchy distributions and Poisson kernel for related concepts.

The inhomogeneous Cauchy – Riemann equations consist of the two equations for a pair of unknown functions u ( x, y ) and v ( x, y ) of two real variables

Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure.

The expected value does not exist for some distributions with large " tails ", such as the Cauchy distribution.

Augustin Louis Cauchy refereed these papers, but refused to accept them for publication for reasons that still remain unclear.

However, in spite of many claims to the contrary, it is widely held that Cauchy recognized the importance of Galois ' work, and that he merely suggested combining the two papers into one in order to enter it in the competition for the Academy's Grand Prize in Mathematics.

As noted before, his first attempt was refused by Cauchy, but in February 1830 following Cauchy's suggestion he submitted it to the Academy's secretary Fourier, to be considered for the Grand Prix of the Academy.

Note that not every probability distribution has a defined mean ; see the Cauchy distribution for an example.

This definition corresponds to the classical definition using Cauchy sequences, except with a constructive twist: for a classical Cauchy sequence, it is required that, for any given distance, there exists ( in a classical sense ) a member in the sequence after which all members are closer together than that distance.