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In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation, is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its particularly simple equidimensional structure, the differential equation can be solved ...
The genius of Cauchy was illustrated in his simple solution of the problem of Apollonius—describing a circle touching three given circles—which he discovered in 1805, his generalization of Euler's formula on polyhedra in 1811, and in several other elegant problems.
If the eigenvalues (the case of Euler equations) are all real the system is defined hyperbolic, and physically eigenvalues represent the speeds of propagation of information. [12] If they are all distinguished, the system is defined strictly hyperbolic (it will be proved to be the case of one-dimensional Euler equations). Furthermore ...
The basic stress analysis problem can be formulated by Euler's equations of motion for continuous bodies (which are consequences of Newton's laws for conservation of linear momentum and angular momentum) and the Euler-Cauchy stress principle, together with the appropriate constitutive equations.
Cauchy boundary conditions are simple and common in second-order ordinary differential equations, ″ = ((), ′ (),), where, in order to ensure that a unique solution () exists, one may specify the value of the function and the value of the derivative ′ at a given point =, i.e.,
In mathematics a Cauchy–Euler operator is a ... It is named after Augustin-Louis Cauchy and Leonhard Euler. The simplest example is ... Cauchy–Euler equation ...
Hence Cauchy-like matrices have a common displacement structure, which can be exploited while working with the matrix. For example, there are known algorithms in literature for approximate Cauchy matrix-vector multiplication with () ops (e.g. the fast multipole method),
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.