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Mathematical series
calculus, a function series is a series where each of its terms is a function, not just a real or complex number. Examples of function series include ordinary
Function_series
Mathematical approximation of a function
analysis, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single
Taylor_series
Formal power series
generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often
Generating_function
Type of function in mathematics
an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at
Analytic_function
Analytic function in mathematics
Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known. The Riemann zeta function ζ(s) is a function of a complex
Riemann_zeta_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the
Sigmoid_function
Special functions of several complex variables
mathematics, theta functions are special functions of several complex variables. Fundamentally, they are a family of continuous functions which encode the
Theta_function
Family of power series in mathematics
hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent
Generalized hypergeometric function
Generalized_hypergeometric_function
Mathematical function
the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial
Beta_function
Functions of an angle
mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of
Trigonometric_functions
Family of solutions to related differential equations
functions. Anger function Bessel polynomials Bessel–Clifford function Bessel–Maitland function Fourier–Bessel series Hahn–Exton q-Bessel function Hankel transform
Bessel_function
Decomposition of periodic functions
A Fourier series (/ˈfʊrieɪ, -iər/) is a series expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example
Fourier_series
Complex-differentiable (mathematical) function
that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is analytic). Holomorphic functions are the central
Holomorphic_function
Function with a repeating pattern
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves
Periodic_function
Association of one output to each input
mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the
Function_(mathematics)
Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2
Error_function
Extension of the factorial function
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic
Gamma_function
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
Special mathematical functions defined on the surface of a sphere
similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines
Spherical_harmonics
Meromorphic function on the complex plane
Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation, is called an L-series. Fundamental
L-function
Arithmetic function related to the divisors of an integer
including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number
Divisor_function
Special mathematical function defined as sin(x)/x
In mathematics, physics and engineering, the sinc function (/ˈsɪŋk/ SINK), denoted by sinc(x), is defined as either sinc ( x ) = sin x x . {\displaystyle
Sinc_function
Solution of a confluent hypergeometric equation
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric
Confluent hypergeometric function
Confluent_hypergeometric_function
Analytic function in mathematics
In analysis, a lacunary function or series is an analytic function that cannot be analytically continued anywhere outside the radius of convergence within
Lacunary_function
Order-preserving mathematical function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept
Monotonic_function
Function studied by Ramanujan
In mathematics, the Ramanujan tau function, studied by Srinivasa Ramanujan, is the function τ : N → Z {\displaystyle \tau :\mathbb {N} \to \mathbb {Z}
Ramanujan_tau_function
Mathematical theorem
{\textstyle \Gamma (s)} is the gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series. Higher-dimensional versions
Ramanujan's_master_theorem
Element of a basis for a function space
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as
Basis_function
Function defined by a hypergeometric series
hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes many other special functions as specific
Hypergeometric_function
Analytic function that does not satisfy a polynomial equation
mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable
Transcendental_function
Functions in mathematics
cosines, functions which are thus referred to as "harmonics." Fourier analysis involves expanding functions on the unit circle in terms of a series of these
Harmonic_function
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Statistical function that defines the quantiles of a probability distribution
probability distribution's quantile function is the inverse of its cumulative distribution function. That is, the quantile function of a distribution D {\displaystyle
Quantile_function
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Functions such that f(–x) equals f(x) or –f(x)
In mathematics, an even function is a real function such that f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} for every x {\displaystyle x} in its domain
Even_and_odd_functions
Power series with negative powers
mathematics, the Laurent series of a complex function f ( z ) {\displaystyle f(z)} is a representation of that function as a power series which includes terms
Laurent_series
Function that is continuous everywhere but differentiable nowhere
infinitely jagged functions (nowadays known as fractal curves). In Weierstrass's original paper, the function was defined as a Fourier series: f ( x ) =
Weierstrass_function
Power series derived from a discrete probability distribution
generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random
Probability generating function
Probability_generating_function
Type of mathematical function
In mathematics, a Dirichlet L-series is a function of the form L ( s , χ ) = ∑ n = 1 ∞ χ ( n ) n s , {\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac
Dirichlet_L-function
Asymmetric sigmoid function
or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). It is a sigmoid function which describes
Gompertz_function
Mathematical transform that expresses a function of time as a function of frequency
takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output
Fourier_transform
Infinite sum of monomials
power series is the Taylor series of some smooth function. In many situations, the center c is equal to zero, for instance for Maclaurin series. In such
Power_series
Type of function in linear algebra
sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm, on a vector space is a real-valued function with
Sublinear_function
Fundamental trigonometric functions
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle:
Sine_and_cosine
Type of mathematical function
Taylor series of an elementary function converges in a neighborhood of every point of its domain. More generally, they are global analytic functions, defined
Elementary_function
Hyperbolic analogues of trigonometric functions
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just
Hyperbolic_functions
Continuous function that is not absolutely continuous
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in
Cantor_function
Mathematical series
Dirichlet series, as are the Dirichlet L-functions. Specifically, the Riemann zeta function ζ(s) is the Dirichlet series of the constant unit function u(n)
Dirichlet_series
Multiplicative function in number theory
The Möbius function μ ( n ) {\displaystyle \mu (n)} is a multiplicative function in number theory introduced by the German mathematician August Ferdinand
Möbius_function
Negative of a convex function
In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to
Concave_function
Mathematical function, inverse of an exponential function
elementary calculus, the series is said to converge to the function ln(z), and the function is the limit of the series. It is the Taylor series of the natural logarithm
Logarithm
Topics referred to by the same term
In mathematics, eta function may refer to: The Dirichlet eta function η(s), a Dirichlet series The Dedekind eta function η(τ), a modular form The Weierstrass
Eta_function
Ratio of polynomial functions
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator
Rational_function
Partial correlation of a time series with its lagged values
In time series analysis, the partial autocorrelation function (PACF) gives the partial correlation of a stationary time series with its own lagged values
Partial autocorrelation function
Partial_autocorrelation_function
Q-analog of hypergeometric series
elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the ratio of
Basic_hypergeometric_series
British period crime drama series
characters" that serve various narrative functions. This situates the protagonists on the margins of society, allowing the series to address themes of intersectionality
Peaky_Blinders_(TV_series)
S-shaped curve
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with the equation f ( x ) = L 1 + e − k ( x − x 0 ) {\displaystyle f(x)={\frac
Logistic_function
Branch of mathematics studying functions of a complex variable
complex function exists. In particular, if a complex function has a derivative, it has derivatives of every order and equals the sum of its Taylor series in
Complex_analysis
Summatory function of the Möbius function
In number theory, the Mertens function is defined for all positive integers n as M ( n ) = ∑ k = 1 n μ ( k ) , {\displaystyle M(n)=\sum _{k=1}^{n}\mu (k)
Mertens_function
Theorem
since power series are infinitely differentiable, so are holomorphic functions (this is in contrast to the case of real differentiable functions), and the
Analyticity of holomorphic functions
Analyticity_of_holomorphic_functions
Number, approximately 3.14
consecutive zeroes of the sine function. The cosine and sine can be defined independently of geometry as a power series, or as the solution of a differential
Pi
Function that is holomorphic on the whole complex plane
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane
Entire_function
Optimization performance test
sphere function was proposed by Kenneth A. De Jong in 1975 as the first item of a series of computational test sets. Because of this, the sphere function is
Sphere_function
Mapping arbitrary data to fixed-size values
A hash function is any function that can be used to map data of arbitrary size to fixed-size values, though there are some hash functions that support
Hash_function
Multivalued function in mathematics
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse
Lambert_W_function
Mathematical relation assigning a probability event to a cost
optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one
Loss_function
Description of limiting behavior of a function
asymptotic analysis for computing function approximations, implicit functions, integrals, iterated functions, series summation, partial sums, solutions
Asymptotic_analysis
Mathematical function with convex lower level sets
In mathematics, a quasiconvex function is a real-valued function defined on a convex subset of a real vector space, such that for any real number y, the
Quasiconvex_function
Limiting a position to an area
Python, the pandas library offers the Series.clip and DataFrame.clip methods. The NumPy library offers the clip function. In the Wolfram Language, it is implemented
Clamp_(function)
Approximation of a function by a polynomial
polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function. The first-order
Taylor's_theorem
Polynomial sequence
nouveau développement en série de fonctions" [On a new development in function series]. C. R. Acad. Sci. Paris (in French). 58: 93–100, 266–273. Collected
Hermite_polynomials
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle
List_of_zeta_functions
Infinite sum approximating a probability distribution in terms of its cumulants
_{n})He_{n}\left({\frac {x-\mu }{\sigma }}\right).} Integrating the series gives us the cumulative distribution function F ( x ) = ∫ − ∞ x f ( u ) d u = Φ ( x ) − ϕ ( x )
Edgeworth_series
Inverse functions of sin, cos, tan, etc.
trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the trigonometric functions, under
Inverse trigonometric functions
Inverse_trigonometric_functions
Mathematical description of quantum state
In quantum mechanics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common
Wave_function
Class of periodic mathematical functions
elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they
Elliptic_function
Point to which functions converge in analysis
mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which
Limit_of_a_function
Mathematical function
{\displaystyle K^{\mu }(x)} to represent these functions. He obtained integral representation and series of functions representations for them. He also established
Conical_function
In mathematics, E-functions are a type of power series that satisfy particular arithmetic conditions on the coefficients. They are of interest in transcendental
E-function
Elementary functions and their finitely iterated integrals
sums. Liouvillian functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841. All elementary functions are Liouvillian. Examples
Liouvillian_function
Method of solution to differential equations
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with
Green's_function
Mathematical concept
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists
Inverse_function
Mathematical function
the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function ϑ(x) or θ(x)
Chebyshev_function
Expression of a function as an infinite sum of simpler functions
In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions. It is a method for calculating
Series_expansion
Number of integers coprime to and less than n
( x ) {\displaystyle \log _{e}(x)} . In number theory, Euler's totient function counts the positive integers up to a given integer n {\displaystyle n}
Euler's_totient_function
Concept in the theory of integral equations
In mathematics, the Liouville–Neumann series is a function series that results from applying the resolvent formalism to solve Fredholm integral equations
Liouville–Neumann_series
Special mathematical function
Li3(z) The polylogarithm function is defined by a power series in z generalizing the Mercator series, which is also a Dirichlet series in s: Li s ( z ) =
Polylogarithm
Type of functions, in mathematical analysis
functions, also known as D-finite functions. When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients
Holonomic_function
American health technology company
Function Health, often stylized as simply Function, is an American company and platform headquartered in Austin, Texas. The venture capital-backed company
Function_Health
Concept in mathematics
Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be
Walsh_function
Types of special mathematical functions
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems
Incomplete_gamma_function
Smooth and compactly supported function
analysis, a bump function is a localized auxiliary function, usually chosen to be smooth and to have compact support. Bump functions are commonly used
Bump_function
Mathematical function
_{1}(z)={\frac {d}{dz}}\psi (z)} where ψ(z) is the digamma function. It may also be defined as the sum of the series ψ 1 ( z ) = ∑ n = 0 ∞ 1 ( z + n ) 2 , {\displaystyle
Trigamma_function
Types of electrical circuits
they all glow. In a series circuit, every device must function for the circuit to be complete. If one bulb burns out in a series circuit, the entire circuit
Series_and_parallel_circuits
Function whose domain is the positive integers
Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum. An arithmetic function a is
Arithmetic_function
Mathematical function with no sudden changes
a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies
Continuous_function
Mathematical function
the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series R i e s z (
Riesz_function
Dark fantasy television series (2005–2020)
"Bedtime Stories". Riverview Hospital in Coquitlam served many functions for the series, including as an asylum in "Asylum", a hospital in "In My Time
Supernatural (American TV series)
Supernatural_(American_TV_series)
Mathematical function
the Riemann–Siegel Z function, the Riemann–Siegel zeta function, the Hardy function, the Hardy Z function and the Hardy zeta function. It can be defined
Z_function
FUNCTION SERIES
FUNCTION SERIES
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
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.
Girl/Female
Bengali, Indian
Fraction of Time
Girl/Female
Afghan, Arabic, Australian, Indian, Muslim
Fiction; Romance; Story
Girl/Female
Tamil
Shrinkhala | à®·à¯à®°à¯€à®¨à¯à®•à®¾à®²à®¾
Born in the month of Shravan, Series
Shrinkhala | à®·à¯à®°à¯€à®¨à¯à®•à®¾à®²à®¾
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.
Girl/Female
Tamil
Chitramala | சிதà¯à®°à®®à®¾à®²à®¾
Series of pictures
Chitramala | சிதà¯à®°à®®à®¾à®²à®¾
Girl/Female
Hindu
Series
Girl/Female
Hindu, Indian
Fraction of the Cosmos
Girl/Female
Tamil
Ankshika | அஂகà¯à®·à¯€à®•à®¾
It’s derived from the root word - anksh that means a fraction. Ankshika means the fraction of the cosmos
Ankshika | அஂகà¯à®·à¯€à®•à®¾
Girl/Female
Hindu
Born in the month of Shravan, Series
Girl/Female
Hindu
Born in the month of Shravan, Series
Surname or Lastname
South German
South German : occupational name for an official in charge of the legal auction of property confiscated in default of a fine; such a sale was known in Middle High German as a gant (from Italian incanto, a derivative of Late Latin inquantare ‘to auction’, from the phrase In quantum? ‘To how much (is the price raised)?’).German : metonymic occupational name for a cooper, from Middle High German ganter, kanter ‘barrel rack’.German : variant of Gander 3.English : occupational name for a glover, from Old French gantier, an agent derivative of gant ‘glove’ (see Gant).
Girl/Female
Tamil
Shrinkhla | à®·à¯à®°à¯€à®¨à¯à®•à®²à®¾
Series
Shrinkhla | à®·à¯à®°à¯€à®¨à¯à®•à®²à®¾
Girl/Female
Tamil
Shrankhla | à®·à¯à®°à®‚கலா
Born in the month of Shravan, Series
Shrankhla | à®·à¯à®°à®‚கலா
Boy/Male
French Greek
Cyrano de Bergerac was a seventeenth-century soldier and science-fiction writer.
Girl/Female
Indian
It’s derived from the root word - anksh that means a fraction. Ankshika means the fraction of the cosmos
Biblical
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Surname or Lastname
English
English : topographic name for someone who lived by a watercourse or road junction, Old English gelǣt, or a habitational name from Leat in Devon, or The Leete in Essex, named with this element.
Boy/Male
Indian
Friction
FUNCTION SERIES
FUNCTION SERIES
Surname or Lastname
English
English : habitational name from places in Hampshire and Sussex called Nutley, from Old English hnutu ‘nut tree’ + lēah ‘(forest) clearing’. The surname has also been established in Ireland since the 17th century.
Girl/Female
Hindu, Indian
Divine Protection
Boy/Male
Indian, Punjabi, Sikh
Happy and Victorious
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Telugu
Lighted; Brighted
Girl/Female
Muslim
Beautiful woman, Distributor, Divider
Boy/Male
Muslim/Islamic
A companion of the Prophet (S.A.W)
Surname or Lastname
English and Irish
English and Irish : variant of Tyrrell.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Lion
Girl/Female
Indian, Tamil
Goddess Amman
Girl/Female
Indian
From Odra.
FUNCTION SERIES
FUNCTION SERIES
FUNCTION SERIES
FUNCTION SERIES
FUNCTION SERIES
a.
Pertaining to, or connected with, a function or duty; official.
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.
n.
The course of action which peculiarly pertains to any public officer in church or state; the activity appropriate to any business or profession.
n.
The act of anointing, smearing, or rubbing with an unguent, oil, or ointment, especially for medical purposes, or as a symbol of consecration; as, mercurial unction.
n.
The act of joining, or the state of being joined; union; combination; coalition; as, the junction of two armies or detachments; the junction of paths.
v. t.
The act of uniting, or the state of being united; junction.
n.
The place or point of union, meeting, or junction; specifically, the place where two or more lines of railway meet or cross.
n.
The act of anointing, or the state of being anointed; unction; specifically (Med.), the rubbing of ointments into the pores of the skin, by which medicinal agents contained in them, such as mercury, iodide of potash, etc., are absorbed.
v. t.
To supply with an organ or organs having a special function or functions.
a.
Pertaining to the function of an organ or part, or to the functions in general.
v. t.
To sell by auction.
n.
A derived function; a function obtained from a given function by a certain algebraic process.
v. t.
To give sanction to; to ratify; to confirm; to approve.
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.
v. i.
Alt. of Functionate
v. t.
To separate by means of, or to subject to, fractional distillation or crystallization; to fractionate; -- frequently used with out; as, to fraction out a certain grade of oil from pretroleum.
n.
The natural or assigned action of any power or faculty, as of the soul, or of the intellect; the exertion of an energy of some determinate kind.
n.
The things sold by auction or put up to auction.
n.
The office, duties, or functions of a minister, servant, or agent; ecclesiastical, executive, or ambassadorial function or profession.
n.
The act of feigning, inventing, or imagining; as, by a mere fiction of the mind.