In mathematics, an elementary function is a function of a single variable that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions.
In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are π and e.
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand.
Liouville's theorem has various meanings, all mathematical results named after Joseph Liouville:
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability.
Joseph Liouville was a French mathematician and engineer.
In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form:
In mathematics, an expression is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations and function composition. Commonly, the allowed functions are nth root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context.
Transcendental number theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space.
In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions.
In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation
In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear partial differential equation satisfied by the conformal factor f of a metric f2(dx2 + dy2) on a surface of constant Gaussian curvature K:
Ahmed I. Zayed is an Egyptian American mathematician. His research interests include Sampling Theory, Wavelets, Medical Imaging, Fractional Fourier transform，Sinc Approximations, Boundary Value Problems, Special Functions and Orthogonal polynomials, Integral transforms.
In applied mathematics, the Atkinson–Mingarelli theorem, named after Frederick Valentine Atkinson and A. B. Mingarelli, concerns eigenvalues of certain Sturm–Liouville differential operators.
The Koopman–von Neumann (KvN) theory is a description of classical mechanics as an operatorial theory similar to quantum mechanics, based on a Hilbert space of complex, square-integrable wavefunctions. As its name suggests, the KvN theory is loosely related to work by Bernard Koopman and John von Neumann in 1931 and 1932, respectively. As explained in this entry, however, the historical origins of the theory and its name are complicated.
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