To solve the PDE we recognize that it is a Cauchy–Euler equation which can be transformed into a diffusion equation by introducing the change-of-variable transformation
Leonhard Euler | Euler | Schrödinger equation | Euler characteristic | Nernst equation | Monge–Ampère equation | Euler Hermes | Boltzmann equation | Diophantine equation | Cauchy distribution | Euler's totient function | Cauchy | Ordinary differential equation | ordinary differential equation | Euler–Maclaurin formula | Euler diagram | Augustin-Louis Cauchy | Young–Laplace equation | wrapped Cauchy distribution | Wave equation | wave equation | Vlasov equation | Tait equation | Smoluchowski coagulation equation | Sauerbrey equation | Redlich–Kwong equation of state | Ramanujan–Nagell equation | Quadratic equation | Prony equation | Pell's equation |
The most general situation in which Cauchy nets apply is Cauchy spaces; these too have a notion of completeness and completion just like uniform spaces.
Namely, such a null sequence becomes an infinitesimal in Cauchy's and Lazare Carnot's terminology.
Then, under Lord North government on Ionian Islands, his talent was remarked and he was sent to study mathematics in Ecole polytechnique, under Biot, Cauchy, Poisson and Fourier.
Halley's method can be viewed as exactly finding the roots of a linear-over-linear Padé approximation to the function, in contrast to Newton's method/Secant method (approximates/interpolates the function linearly) or Cauchy's method/Muller's method (approximates/interpolates the function quadratically).
The word "holomorphic" was introduced by two of Cauchy's students, Briot (1817–1882) and Bouquet (1819–1895), and derives from the Greek ὅλος (holos) meaning "entire", and μορφή (morphē) meaning "form" or "appearance".
Also in 1974 he made the first observation of sub-Poissonian statistics for light (via a violation of the Cauchy–Schwarz inequality for classical electromagnetic fields), and thereby, for the first time, demonstrated an unambiguous particle-like character for photons.
Since the right-hand side of the identity is clearly non-negative, it implies Cauchy's inequality in the finite-dimensional real coordinate space ℝn and its complex counterpart ℂn.
In Bayesian statistics, the log-Cauchy distribution can be used to approximate the improper Jeffreys-Haldane density, 1/k, which is sometimes suggested as the prior distribution for k where k is a positive parameter being estimated.
Cauchy distribution, also known as the Lorentz distribution, Lorentzian function, or Cauchy–Lorentz distribution
A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution.
The wrapped Cauchy distribution is often found in the field of spectroscopy where it is used to analyze diffraction patterns (e.g. see Fabry–Pérot interferometer)