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Mathematical function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane
Dedekind_eta_function
Topics referred to by the same term
theory, Dedekind function can refer to any of three functions, all introduced by Richard Dedekind Dedekind eta function Dedekind psi function Dedekind zeta
Dedekind_function
Generalization of the Riemann zeta function for algebraic number fields
mathematics, the Dedekind zeta function of an algebraic number field K, usually denoted ζ K ( s ) {\displaystyle \zeta _{K}(s)} , is an analytic function that represents
Dedekind_zeta_function
German mathematician (1831–1916)
Richard Dedekind Dedekind cut Dedekind domain Dedekind eta function Dedekind-infinite set Dedekind number Dedekind psi function Dedekind sum Dedekind zeta
Richard_Dedekind
Arithmetical function
In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by ψ ( n ) = n ∏ p | n ( 1 + 1 p ) , {\displaystyle
Dedekind_psi_function
functions Theta functions Neville theta functions Modular lambda function Closely related are the modular forms, which include J-invariant Dedekind eta
List of mathematical functions
List_of_mathematical_functions
Set with an equinumerous proper subset
that there exists a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection
Dedekind-infinite_set
Combinatorial sequence of numbers
mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number M (
Dedekind_number
Meromorphic function on the complex plane
now known as the Riemann zeta function). Most notably, the mathematicians Bernhard Riemann (1826-1866), Richard Dedekind (1831-1916), Erich Hecke (1887-1947)
L-function
Type of Dirichlet series associated to number field extensions
theory and are generalizations of better-known functions like Dedekind zeta functions or Dirichlet L-functions. Some of their expected properties turned out
Artin_L-function
In mathematics, Dedekind sums are certain finite sums of products of a sawtooth function. Dedekind introduced them in the 1880's to express the functional
Dedekind_sum
Number of partitions of an integer
specifically the Dedekind eta function. The same sequence of pentagonal numbers appears in a recurrence relation for the partition function: p ( n ) = ∑ k
Partition function (number theory)
Partition_function_(number_theory)
axiom Dedekind completeness Dedekind cut Dedekind discriminant theorem Dedekind domain Dedekind eta function Dedekind function Dedekind group Dedekind number
List of things named after Richard Dedekind
List_of_things_named_after_Richard_Dedekind
Algebra with unique prime factorization
In mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into
Dedekind_domain
Number of integers coprime to and less than n
product of the first 120569 primes. Carmichael function (λ) Dedekind psi function (𝜓) Divisor function (σ) Duffin–Schaeffer conjecture Generalizations
Euler's_totient_function
Order-preserving mathematical function
b) or (a and c) or (b and c)). The number of such functions on n variables is known as the Dedekind number of n. SAT solving, generally an NP-hard task
Monotonic_function
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
Analytic function in mathematics
the Dirichlet L-functions and the Dedekind zeta function. For other related functions see the articles zeta function and L-function. The polylogarithm
Riemann_zeta_function
Set whose pairs have minima and maxima
y} have the same length, then the lattice is said to satisfy the Jordan–Dedekind chain condition. A lattice ( L , ≤ ) {\displaystyle (L,\leq )} is called
Lattice_(order)
Mathematical conjecture about zeros of L-functions
they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Axioms for the natural numbers
mathematical logic, the Peano axioms (/piˈɑːnoʊ/; [peˈaːno]), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers
Peano_axioms
Special functions of several complex variables
{1}{2i}}(1-2x)\log q,{\frac {1}{q}}\right).} Let η(τ) be the Dedekind eta function, and the argument of the theta function as the nome q = eπiτ. Then, θ 2 ( q ) = ϑ 10
Theta_function
Topics referred to by the same term
Psi function can refer, in mathematics, to the ordinal collapsing function ψ ( α ) {\displaystyle \psi (\alpha )} the Dedekind psi function ψ ( n ) {\displaystyle
Psi_function
relation Cyclotomic polynomials H. G. Dawson: Dawson function Richard Dedekind: Dedekind eta function Charles F. Dunkl: Dunkl operator, Jacobi–Dunkl operator
List of eponyms of special functions
List_of_eponyms_of_special_functions
zeta function of a dynamical system Barnes zeta function or double zeta function Beurling zeta function of Beurling generalized primes Dedekind zeta function
List_of_zeta_functions
Mathematician (1845–1918)
mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on holiday in Gersau in Switzerland
Georg_Cantor
Number representing a continuous quantity
real functions and real-valued sequences. One modern axiomatic definition is that real numbers form the unique (up to an isomorphism) Dedekind-complete
Real_number
Class of mathematical functions
{\displaystyle \eta } is the Dedekind eta function. For the Fourier coefficients of Δ {\displaystyle \Delta } , see Ramanujan tau function. e 1 {\displaystyle
Weierstrass_elliptic_function
"Continued Fractions and Modular Functions". The function η ( τ ) {\displaystyle \eta (\tau )} is the Dedekind eta function and ( e 2 π i τ ) α {\displaystyle
Weber_modular_function
Conjecture on zeros of the zeta function
extends the Riemann hypothesis to all Dedekind zeta functions of algebraic number fields. Since Dedekind zeta function for abelian extension of the rationals
Riemann_hypothesis
abstract algebra, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains
Dedekind–Hasse_norm
Function studied by Ramanujan
where ϕ {\displaystyle \phi } is the Euler function, η {\displaystyle \eta } is the Dedekind eta function, Δ ( z ) {\displaystyle \Delta (z)} is the modular
Ramanujan_tau_function
About mathematical functions
himself, as in previous conceptions, to real (or complex) functions, Dedekind defines a function as a single-valued mapping between any two sets: What was
History of the function concept
History_of_the_function_concept
Analytic function on the upper half-plane with a certain behavior under the modular group
The function ε ( a , b , c , d ) {\displaystyle \varepsilon (a,b,c,d)} is called the nebentypus of the modular form. Functions such as the Dedekind eta
Modular_form
Arithmetical function
{\displaystyle J_{k}(n)\sim {\frac {n^{k}}{\zeta (k+1)}}} . The Dedekind psi function is ψ ( n ) = J 2 ( n ) J 1 ( n ) {\displaystyle \psi (n)={\frac
Jordan's_totient_function
Finite collection of distinct objects
powerset of the powerset of S {\displaystyle S} is Dedekind-finite (see below). Every surjective function from ℘ ( ℘ ( S ) ) {\displaystyle \wp {\bigl (}\wp
Finite_set
Theorem in set theory
Ernst Zermelo discovered Dedekind's proof and in 1908 he publishes his own proof based on the chain theory from Dedekind's paper Was sind und was sollen
Schröder–Bernstein_theorem
Size of a possibly infinite set
or |Y| ≤ |X|. A set X is called Dedekind-infinite if there exists a proper subset Y of X with |X| = |Y|, and Dedekind-finite if such a subset does not
Cardinal_number
Property of a partially ordered set
completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness. It can be used to prove many of the fundamental results of
Least-upper-bound_property
Mathematical identities related to integer partitions
{1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}} The Dedekind eta function identities for the functions G and H result by combining only the following two
Rogers–Ramanujan_identities
Modular function in mathematics
)=g_{2}(\tau )^{3}-27g_{3}(\tau )^{2}=(2\pi )^{12}\,\eta (\tau )^{24}} , Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )} , and modular invariants, g
J-invariant
Formula in number theory
invariants of an algebraic number field to a special value of its Dedekind zeta function. We start with the following data: K is a number field. [K : Q]
Class_number_formula
Process of repeating items in a self-similar way
F(n+1)=f(F(n))} for any natural number n. Dedekind was the first to pose the problem of unique definition of set-theoretical functions on N {\displaystyle \mathbb {N}
Recursion
Proof in set theory
thereof. Various models have been studied, such as the Cauchy reals or the Dedekind reals, among others. The former relate to quotients of sequences while
Cantor's_diagonal_argument
Shape taken by a self-gravitating fluid body rotating at constant velocity
the Dedekind ellipsoid maintains a fixed orientation, with the constituent fluid circulating within it. This is a direct consequence of Dedekind's theorem
Jacobi_ellipsoid
Function whose domain is the positive integers
{\displaystyle c_{q}(n)c_{r}(n)=c_{qr}(n).} The Dedekind psi function, used in the theory of modular functions, is defined by the formula ψ ( n ) = n ∏ p |
Arithmetic_function
Number used for counting
Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of Dedekind's axioms in
Natural_number
Type of zeta function
generalizes the Riemann zeta function and Dedekind zeta function to higher dimensions. The arithmetic zeta function is one of the most fundamental objects
Arithmetic_zeta_function
Mathematical functions
11995279. JSTOR 2321821. Roy, Ranjan (2017). Elliptic and Modular Functions from Gauss to Dedekind to Hecke. Cambridge University Press. p. 28. ISBN 978-1-107-15938-9
Lemniscate_elliptic_functions
Smallest complete lattice containing a partial order
constructed it, and after Richard Dedekind because its construction generalizes the Dedekind cuts used by Dedekind to construct the real numbers from
Dedekind–MacNeille_completion
Mathematical concept
poles. More generally, the Riemann zeta function and the L-series can be replaced by the Dedekind zeta function of an algebraic number field or a Hecke
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Type of generalization of periodic functions in Euclidean space
harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G {\displaystyle G} to the complex numbers (or
Automorphic_form
Finite extension of the rationals
equation for the zeta-function are needed to define the function for all s). The Dedekind zeta-function generalizes the Riemann zeta-function in that ζ Q {\displaystyle
Algebraic_number_field
Mathematical functions related to Weierstrass's elliptic function
Weierstrass eta function should not be confused with either the Dedekind eta function or the Dirichlet eta function. The Weierstrass p-function is related
Weierstrass_functions
Evaluates a certain product of values of the Gamma function at rational values
certain product of values of the gamma function at rational values in terms of values of the Dedekind eta function at imaginary quadratic irrational numbers
Chowla–Selberg_formula
Mathematical function
Euler function, which is closely related to the Dedekind eta function. The Jacobi theta function may be written in terms of the Ramanujan theta function as:
Ramanujan_theta_function
Branch of mathematics that studies sets
discontinuous functions. A young Georg Cantor entered into this area, which led him to the study of point-sets. Around 1871, influenced by Riemann, Dedekind began
Set_theory
One-to-one correspondence
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the
Bijection
Axiom of set theory
implies the equivalence of infinite and Dedekind-infinite sets, but that the equivalence of infinite and Dedekind-infinite sets does not imply the axiom
Axiom_of_choice
Type of character in number theory
to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have
Hecke_character
Mathematical function
{\displaystyle (3n^{2}-n)/2} is a pentagonal number. The Euler function is related to the Dedekind eta function as ϕ ( e 2 π i τ ) = e − π i τ / 12 η ( τ ) . {\displaystyle
Euler_function
Projective variety that is also an algebraic group
field of a Dedekind domain, for any nonzero prime of your Dedekind domain, there is a map from the Dedekind domain to the quotient of the Dedekind domain
Abelian_variety
Function computable with bounded loops
In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all
Primitive_recursive_function
Collection of mathematical objects
symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Mathematics typically does not define precisely what
Set_(mathematics)
German mathematician (1805–1859)
close contact with the new generation of researchers, especially Richard Dedekind and Bernhard Riemann. After moving to Göttingen he was able to obtain a
Peter Gustav Lejeune Dirichlet
Peter_Gustav_Lejeune_Dirichlet
Nonexistence of gaps in the number line
real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts
Completeness of the real numbers
Completeness_of_the_real_numbers
On the distribution of prime numbers
of methods using the Riemann zeta function to estimate distribution of primes in integers to Dedekind zeta functions, and to use them for distribution
Hilbert's_eighth_problem
Mathematical theorem about the real analytic Eisenstein series
a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more complicated
Kronecker_limit_formula
Type of lattice in mathematical order theory
semimodularity. Modular lattices are sometimes called Dedekind lattices after Richard Dedekind, who discovered the modular identity in several motivating
Modular_lattice
Seventh letter in the Greek alphabet
lambda calculus. Mathematics, the Dirichlet eta function, Dedekind eta function, and Weierstrass eta function. In category theory, the unit of an adjunction
Eta
Concept in abstract algebra
to the integers under addition. R {\displaystyle R} is a local ring, a Dedekind domain, and not a field. R {\displaystyle R} is Noetherian and a local
Discrete_valuation_ring
German mathematician (1826–1866)
Grunde liegen. It was not published until twelve years later in 1868 by Dedekind, two years after his death. Its early reception appears to have been slow
Bernhard_Riemann
Branch of number theory
corresponds to the Riemann zeta function. When K is a Galois extension, the Dedekind zeta function is the Artin L-function of the regular representation
Algebraic_number_theory
Paradox in set theory
a contradiction (to Cantor's theorem), as he told Hilbert and Richard Dedekind by letter. Hilbert also formulated his own paradox, which relied on reasoning
Russell's_paradox
Natural number
Ramanujan τ {\displaystyle \tau } -function and which is (up to a constant multiplier) the 24th power of the Dedekind eta function: Δ ( τ ) = ( 2 π ) 12 η 24
12_(number)
Set whose elements all belong to another set
Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset · Superset Transitive Uncountable
Subset
Symmetric holomorphic function
(\tau )=k^{2}(\tau )} . In terms of the Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )} and theta functions, λ ( τ ) = ( 2 η ( τ 2 ) η 2 ( 2 τ )
Modular_lambda_function
Mathematical function
In mathematics, Hooley's delta function ( Δ ( n ) {\displaystyle \Delta (n)} ), also called Erdős--Hooley delta-function, defines the maximum number of
Hooley's_delta_function
Function in algebra
ring is RP. The previous example can be generalized to Dedekind domains. Let R be a Dedekind domain, K its field of fractions, and let P be a non-zero
Valuation_(algebra)
First article on transfinite set theory
Schwarz. Cantor's failure to mention Dedekind's contributions damaged his relationship with Dedekind. Dedekind stopped replying to his letters and did
Cantor's first set theory article
Cantor's_first_set_theory_article
Collection of sets in mathematics that can be defined based on a property of its members
"classes". In ZF, the concept of a function can also be generalised to classes. A class function is not a function in the usual sense, since it is not
Class_(set_theory)
Concept in mathematics
function exists without constructing it. As an example of an application of ACω, here is a proof (from ZF + ACω) that every infinite set is Dedekind-infinite:
Axiom_of_countable_choice
Infinite set that is not countable
incomparable to ℵ 0 {\displaystyle \aleph _{0}} (namely, the cardinalities of Dedekind-finite infinite sets). Sets of these cardinalities satisfy the first three
Uncountable_set
Series related to Ramanujan's pi formulas
\end{aligned}}} with the j-function j(τ), Eisenstein series E4, and Dedekind eta function η(τ). The first expansion is the McKay–Thompson
Ramanujan–Sato_series
Iterative algorithm on numbers
sequence. Repeat step 2. The sequence is called a Kaprekar sequence and the function K b ( n ) = α − β {\displaystyle K_{b}(n)=\alpha -\beta } is the Kaprekar
Kaprekar's_routine
Mathematical set formed from two given sets
as simply ×Xi. If f is a function from X to A and g is a function from Y to B, then their Cartesian product f × g is a function from X × Y to A × B with
Cartesian_product
Natural number
4007/annals.2017.185.3.8. Apostol, Tom M. (1990). "The Dedekind eta function". Modular Functions and Dirichlet Series in Number Theory. Graduate Texts
24_(number)
American mathematician
to special values of the Riemann zeta function. Zagier found a formula for the value of the Dedekind zeta function of an arbitrary number field at s = 2
Don_Zagier
Mathematical set that can be enumerated
which are incomparable to N {\displaystyle \mathbb {N} } , the so-called Dedekind finite infinite sets. In 1874, in his first set theory article, Cantor
Countable_set
Algebraic structure with addition, multiplication, and division
algebraic numbers (which is a field in Dedekind's sense), but on the other hand was more abstract than Dedekind's in that it made no specific assumption
Field_(mathematics)
Axiomatic definition of a class of L-functions
of the zeta function, like Dirichlet L-functions or Dedekind zeta functions, belong to the Selberg class. Examples of primitive functions include the
Selberg_class
Algebraic structure with addition and multiplication
contributions by Richard Dedekind, David Hilbert, Abraham Fraenkel, and Emmy Noether. Rings were first formalized as a generalization of Dedekind domains that occur
Ring_(mathematics)
Extremely small quantity in calculus; thing so small that there is no way to measure it
d'Analyse, and in defining an early form of a Dirac delta function. As Cantor and Dedekind were developing more abstract versions of Stevin's continuum
Infinitesimal
Mathematical set containing no elements
exists precisely one function f {\displaystyle f} from ∅ {\displaystyle \varnothing } to A , {\displaystyle A,} the empty function. As a result, the empty
Empty_set
Arithmetic operation
function xy has no limit at the point (0, 0). One may consider at what points this function does have a limit. More precisely, consider the function f
Exponentiation
Decomposition of periodic functions
periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum
Fourier_series
Set that is not a finite set
a set is infinite if and only if the power set of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself. If the axiom
Infinite_set
Cantor–Bendixson rank Cantor–Bendixson theorem Cantor–Bernstein theorem Cantor–Dedekind axiom Heine–Cantor theorem Cantor–Schröder–Bernstein theorem Cantor–Schröder–Bernstein
List of things named after Georg Cantor
List_of_things_named_after_Georg_Cantor
3-volume treatise on mathematics, 1910–1913
[logical-] function, and, if φx ≡ ψx, the substitution of φx for p in a [logical-] function gives the same truth-value to the truth-function as the substitution
Principia_Mathematica
Unsolved problem in mathematics
is the Dedekind eta function. Δ ( z ) {\textstyle \Delta (z)} is a holomorphic cusp form of weight 12 and level 1. The Ramanujan tau function τ ( n )
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
DEDEKIND FUNCTION
DEDEKIND FUNCTION
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, a great 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.
Male
Egyptian
, a high Egyptian functionary.
Male
Celtic
, great justiciary, or functionary.
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.
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
Egyptian
, Functionary of the Interior.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, the son of the functionary Heknofre.
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.
Biblical
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DEDEKIND FUNCTION
DEDEKIND FUNCTION
Girl/Female
English American German Teutonic
Famous.
Boy/Male
Hindu
Another name of Lord Shiva, Greatest God
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Lord Ganesha
Girl/Female
English
Blend of Geri plus Erica.
Girl/Female
Latin Spanish
Small.
Boy/Male
English
From the elves'valley.
Girl/Female
Latin
Beloved.
Girl/Female
Arabic, Australian, French, Muslim
Aspiration; Hope
Boy/Male
Indian, Punjabi, Sikh, Traditional
The Vessel of Purity and Righteousness of the Spiritual Family
Boy/Male
Indian, Sanskrit
Pure; Clean
DEDEKIND FUNCTION
DEDEKIND FUNCTION
DEDEKIND FUNCTION
DEDEKIND FUNCTION
DEDEKIND FUNCTION
pl.
of Functionary
n.
Fig.: Any cavity, or hollow place, in which any function may be conceived of as operating.
a.
Of or pertaining to the vessels of animal and vegetable bodies; as, the vascular functions.
n.
One deputed or authorized to perform the functions of another; a substitute in office; a deputy.
prep.
Acting as a substitute; -- said of abnormal action which replaces a suppressed normal function; as, vicarious hemorrhage replacing menstruation.
v. i.
Alt. of Functionate
v. i.
To execute or perform a function; to transact one's regular or appointed business.
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.
Destitute of function, or of an appropriate organ. Darwin.
a.
Belonging or relating to life, either animal or vegetable; as, vital energies; vital functions; vital actions.
a.
Pertaining to, or connected with, a function or duty; official.
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.
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.
a.
Pertaining to the function of an organ or part, or to the functions in general.
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.
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. t.
To assign to some function or office.
adv.
In a functional manner; as regards normal or appropriate activity.
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.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.