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In optics, Cauchy's transmission equation is an empirical relationship between the refractive index and wavelength of light for a particular transparent material. It is named for the mathematician Augustin-Louis Cauchy , who originally defined it in 1830 in his article "The refraction and reflection of light".
As written in the Cauchy momentum equation, the stress terms p and τ are yet unknown, so this equation alone cannot be used to solve problems. Besides the equations of motion—Newton's second law—a force model is needed relating the stresses to the flow motion. [12]
Nevertheless, if a function satisfies the Cauchy–Riemann equations in an open set in a weak sense, then the function is analytic. More precisely: [ 18 ] If f ( z ) is locally integrable in an open domain Ω ∈ C , {\displaystyle \Omega \in \mathbb {C} ,} and satisfies the Cauchy–Riemann equations weakly, then f agrees almost everywhere ...
The elastic wave equation (also known as the Navier–Cauchy equation) in three dimensions describes the propagation of waves in an isotropic homogeneous elastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials.
The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution.It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution.
Cauchy's functional equation is the functional equation: (+) = + (). A function that solves this equation is called an additive function.Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely : for any rational constant .
The next three years Cauchy was mainly on unpaid sick leave; he spent his time fruitfully, working on mathematics (on the related topics of symmetric functions, the symmetric group and the theory of higher-order algebraic equations). He attempted admission to the First Class of the Institut de France but failed on three different occasions ...
However, according to Cauchy's fundamental theorem, [11] also called Cauchy's stress theorem, [1] merely by knowing the stress vectors on three mutually perpendicular planes, the stress vector on any other plane passing through that point can be found through coordinate transformation equations. Cauchy's stress theorem states that there exists ...