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Elementary functions and their finitely iterated integrals
mathematics, the Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be
Liouvillian_function
Multivalued function in mathematics
elementary (Liouvillian) functions, the first published proof did not appear until 2008. There are countably many branches of the W function, denoted by
Lambert_W_function
Type of mathematical function
most special functions are not elementary. Non-elementary functions include: the gamma function non-elementary Liouvillian functions, including the
Elementary_function
Mathematical formula involving a given set of operations
approximationsPages displaying short descriptions of redirect targets Liouvillian function – Elementary functions and their finitely iterated integrals Symbolic regression –
Closed-form_expression
Criterion for integration in terms of elementary functions
Elementary function – Type of mathematical function Elementary function arithmetic – System of arithmetic in proof theory Liouvillian function – Elementary
Liouville's theorem (differential algebra)
Liouville's_theorem_(differential_algebra)
operator Liouville space Liouville surface Liouville–Neumann series Liouvillian function Riemann–Liouville integral Quantum Liouville equation Sturm–Liouville
List of things named after Joseph Liouville
List_of_things_named_after_Joseph_Liouville
Type of mathematical function
) {\displaystyle (-\pi ,\pi )} . In fact, all elementary functions and Liouvillian functions are Pfaffian. Consider the structure R = ( R , + , − , ⋅
Pfaffian_function
Study of differential field extensions induced by linear differential equations
equation y′ = By.) An extension F ⊆ K of differential fields is called Liouvillian if all constants are in F, and K can be generated by adjoining a finite
Picard–Vessiot_theory
Integrals not expressible in closed-form from elementary functions
of the set of the elementary functions is the set of the Liouvillian functions. Algebraic function – Mathematical function Closed-form expression – Mathematical
Nonelementary_integral
Key result in Hamiltonian mechanics and statistical mechanics
}{\partial t}}=\{H,\rho \}} or, in terms of the linear Liouville operator or Liouvillian, i L ^ = ∑ i = 1 n [ ∂ H ∂ p i ∂ ∂ q i − ∂ H ∂ q i ∂ ∂ p i ] = − { H
Liouville's theorem (Hamiltonian)
Liouville's_theorem_(Hamiltonian)
Markovian quantum master equation for density matrices (mixed states)
(GKSL) master equation, master equation in Lindblad form, quantum Liouvillian, or Lindbladian is one of the general forms of Markovian master equations
Lindbladian
Operation in Hamiltonian mechanics
i{\hat {L}}=-\{{\mathcal {H}},\cdot \}} , is sometimes referred to as the Liouvillian (see Liouville's theorem (Hamiltonian)). The concept of Poisson brackets
Poisson_bracket
Formulation of classical mechanics in terms of Hilbert spaces
}{\partial t}}\rho (x,p,t)={\hat {L}}\rho (x,p,t)} with the self-adjoint Liouvillian L ^ = − i ∂ H ( x , p ) ∂ p ∂ ∂ x + i ∂ H ( x , p ) ∂ x ∂ ∂ p , {\displaystyle
Koopman–von Neumann classical mechanics
Koopman–von_Neumann_classical_mechanics
Application of information theory to thermodynamics and statistical mechanics
respectively specified particular models for the underlying state space (e.g. Liouvillian phase space); and by respectively specified particular partial descriptions
Maximum entropy thermodynamics
Maximum_entropy_thermodynamics
constant. Whittaker–Hill equation Ince–Gaussian beam Cheung, Tsz Yung. "Liouvillian solutions of Whittaker-Ince equation". Journal of Symbolic Computation
Ince_equation
State of a physical system in phase space
Hamiltonian for a closed quantum system or the eigenoperator of the Liouvillian for an open quantum system. For a many-body system, phase space crystal
Phase_space_crystal
Swedish physicist (1916–2000)
Löwdin, Per-Olov (1982). "On operators, superoperators, hamiltonians, and liouvillians". International Journal of Quantum Chemistry. 22 (S16): 485–560. doi:10
Per-Olov_Löwdin
LIOUVILLIAN FUNCTION
LIOUVILLIAN FUNCTION
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, Functionary of the Interior.
Male
Egyptian
, a great functionary.
Male
Egyptian
, a high Egyptian functionary.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, the son of the functionary Heknofre.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Male
Celtic
, great justiciary, or functionary.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Biblical
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LIOUVILLIAN FUNCTION
LIOUVILLIAN FUNCTION
Boy/Male
Greek Latin
A judge in the underworld.
Girl/Female
Hindu, Indian
Learned
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Lord Swaminarayan; Lord Shiva
Girl/Female
Welsh
Jewel of the sea.
Boy/Male
Hindu
A name of Lord Shiva
Girl/Female
Spanish
Certain fortune; fate.
Boy/Male
French
Chubby cheeks.
Boy/Male
British, Gujarati, Hindu, Indian
Best
Boy/Male
Australian, British, Dutch, English, French, German, Hebrew, Italian, Latin, Swedish
Supplanter; Held by the Heel
Male
Spanish
Spanish form of Latin Martinus, MARTÃN means "of/like Mars."
LIOUVILLIAN FUNCTION
LIOUVILLIAN FUNCTION
LIOUVILLIAN FUNCTION
LIOUVILLIAN FUNCTION
LIOUVILLIAN FUNCTION
v. t.
To assign to some function or office.
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
pl.
of Functionary
n.
The doctrine that all the functions of a living organism are due to an unknown vital principle distinct from all chemical and physical forces.
n.
A certain function relating to a system of forces and their points of application, -- first used by Clausius in the investigation of problems in molecular physics.
v. i.
To execute or perform a function; to transact one's regular or appointed business.
prep.
Acting as a substitute; -- said of abnormal action which replaces a suppressed normal function; as, vicarious hemorrhage replacing menstruation.
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
a.
Pertaining to the function of an organ or part, or to the functions in general.
a.
Pertaining to, or connected with, a function or duty; official.
n.
One deputed or authorized to perform the functions of another; a substitute in office; a deputy.
a.
Belonging or relating to life, either animal or vegetable; as, vital energies; vital functions; vital actions.
n.
Fig.: Any cavity, or hollow place, in which any function may be conceived of as operating.
a.
Destitute of function, or of an appropriate organ. Darwin.
a.
Of, pertaining to, or designating, certain secret tribunals which flourished in Germany from the end of the 12th century to the middle of the 16th, usurping many of the functions of the government which were too weak to maintain law and order, and inspiring dread in all who came within their jurisdiction.
adv.
In a functional manner; as regards normal or appropriate activity.
v. i.
Alt. of Functionate
a.
Having relation to growth or nutrition; partaking of simple growth and enlargement of the systems of nutrition, apart from the sensorial or distinctively animal functions; vegetal.
a.
Of or pertaining to the vessels of animal and vegetable bodies; as, the vascular functions.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.