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Mathematical function on a space that is invariant under the action of some group
mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient
Automorphic_function
Type of generalization of periodic functions in Euclidean space
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G {\displaystyle G} to the complex numbers
Automorphic_form
Mathematical concept
In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic representation π of a reductive
Automorphic_L-function
Conjectures connecting number theory and geometry
of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by the Canadian mathematician Robert
Langlands_program
Number whose square ends in the same digits
{\displaystyle k} digits is an automorphic number if n {\displaystyle n} is a fixed point of the polynomial function f ( x ) = x 2 {\displaystyle f(x)=x^{2}}
Automorphic_number
Topics referred to by the same term
mathematics Automorphic form, in mathematics Automorphic representation, in mathematics Automorphic L-function, in mathematics Automorphism, in mathematics
Automorphic
Function defined by a hypergeometric series
is positive, zero or negative; and the s-maps are inverse functions of automorphic functions for the triangle group 〈p, q, r〉 = Δ(p, q, r). The monodromy
Hypergeometric_function
Analytic function on the upper half-plane with a certain behavior under the modular group
growth condition. A modular form is a special case of an automorphic form, which are functions defined on Lie groups that transform nicely with respect
Modular_form
Representation Theory and Automorphic Functions. Boston: Academic Press. ISBN 0-12-279506-7. Bump, Daniel (1998). Automorphic Forms and Representations
Schwartz–Bruhat_function
Concept in information theory
chain model, the Rényi entropy as a function of α can be calculated explicitly because it is an automorphic function with respect to a particular subgroup
Rényi_entropy
Mathematical conjectures in class field theory
Armand (1979), "Automorphic L-functions", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure
Local_Langlands_conjectures
Mathematical concept
equivalence between motivic and automorphic L-functions postulated in the Langlands program can be tested. Automorphic forms realized in the cohomology
Shimura_variety
German mathematician (1849–1925)
especially equations he invented, satisfied by elliptic modular functions and automorphic functions. Klein showed that the modular group moves the fundamental
Felix_Klein
Japanese mathematician (1930–2019)
equivalence between motivic and automorphic L-functions postulated in the Langlands program could be tested: automorphic forms realized in the cohomology
Goro_Shimura
Kind of complex manifold
condition. These are automorphic functions, more precisely, the automorphic functions used in the transformation laws for theta functions. Also, any such map
Complex_torus
Type of Dirichlet series associated to number field extensions
the complex-analytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and the Langlands program. So far
Artin_L-function
36 mathematical problems stated in 1955
this form is an elliptic differential of the field of associated automorphic functions. Now, going through these observations backward, is it possible
Taniyama's_problems
British mathematician and historian of science (1873–1956)
a century. Throughout his career, he wrote papers on automorphic functions and special functions in pure mathematics as well as on electromagnetism, general
E._T._Whittaker
Distance from zero to a number
Press. ISBN 0-12-622760-8. Siegel, Carl Ludwig (1942). "Note on automorphic functions of several variables". Annals of Mathematics. Second Series. 43
Absolute_value
23 mathematical problems stated in 1900
monodromy group. 22. Uniformization of analytic relations by means of automorphic functions. 23. Further development of the methods of the calculus of variations
Hilbert's_problems
Special functions of several complex variables
the theta series to automorphic forms with respect to arbitrary Fuchsian groups. In the following, three important theta function values are to be derived
Theta_function
German mathematician (1826–1866)
and poles) of a Riemann surface. According to Detlef Laugwitz, automorphic functions appeared for the first time in an essay about the Laplace equation
Bernhard_Riemann
Unsolved problem in mathematics
more generally, automorphic forms. The name of the conjecture comes from Srinivasa Ramanujan, who proposed it for Ramanujan tau function, and Hans Petersson
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Arithmetic Theory of Automorphic Functions, Princeton University Press, 1994. ISBN 0-691-08092-5 Gelbart, Stephen, Automorphic Forms on Adele Groups
Cusp_form
classes of Q-rational parabolic subgroups one should construct automorphic functions (from objects from spaces of lower dimensions) whose constant terms
Parabolic_induction
German mathematician (1911–1992)
was also concerned with automorphic functions in several variables, Siegel modular functions, and associated zeta functions. Maass, Hans (1949), "Über
Hans_Maass
MR 0088511 Venkov, A. B. Spectral theory of automorphic functions. Proc. Steklov. Inst. Math, 1982. Sunada, T., L-functions in geometry and some applications,
Selberg_zeta_function
Russian mathematician
doctorate (higher doctoral degree) with dissertation Spectral theory of automorphic functions (Russian). He was a visiting scholar at IHES, at the University
Alexei_Venkov
Meromorphic function on the complex plane
function, π {\displaystyle \textstyle \pi } denotes the automorphic number, and d {\displaystyle \textstyle d} denotes the degree of the L-function mentioned
L-function
In mathematics, an automorphic factor is a certain type of analytic function, defined on subgroups of SL(2,R), appearing in the theory of modular forms
Automorphic_factor
Completes the Langlands program for general linear groups over algebraic function fields
program for general linear groups over algebraic function fields, by giving a correspondence between automorphic forms on these groups and representations of
Lafforgue's_theorem
acting on a domain D and H(z) is any meromorphic function on D, then one obtains an automorphic function by averaging over Γ: ∑ γ ∈ Γ H ( γ ( z ) ) . {\displaystyle
Poincaré series (modular form)
Poincaré_series_(modular_form)
Israeli mathematician (1929–2009)
geometry. His main contribution and impact was in the area of automorphic forms and L-functions. For the last 30 years of his life he suffered from Parkinson's
Ilya_Piatetski-Shapiro
On uniformization of analytic relations
It entails the uniformization of analytic relations by means of automorphic functions. The entirety of the original problem statement is as follows: As
Hilbert's twenty-second problem
Hilbert's_twenty-second_problem
Mathematical formula in harmonic analysis
been a standard tool for studying analytic properties of automorphic forms and their L-functions. There have been numerous results coming out the Voronoi
Voronoi_formula
Mathematical theorem
theory of automorphic forms and in analytic number theory. The trace formula is also central to the analytic theory of the Selberg zeta function. It can
Selberg_trace_formula
Mathematical structure
applies to elliptic curves and elliptic functions; and in fact to curves of genus > 1 and automorphic functions.) The properties of algebraic curves, such
Transcendental_curve
representations of L-functions, is a technique for directly constructing and analytically continuing several important examples of automorphic L-functions. Some authors
Rankin–Selberg_method
Arithmetical function
Weisstein, Eric W. "Dedekind Function". MathWorld. Goro Shimura (1971). Introduction to the Arithmetic Theory of Automorphic Functions. Princeton. (page 25,
Dedekind_psi_function
Mathematical concept
In mathematics, the term standard L-function refers to a particular type of automorphic L-function described by Robert P. Langlands. Here, standard refers
Standard_L-function
Axiomatic definition of a class of L-functions
class is equal to class of automorphic L-functions. Primitive functions are expected to be associated with irreducible automorphic representations. It is
Selberg_class
Mathematical functions that quantify complexity
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of
Height_function
Algebraic variety
Shimura, Goro (1994) [1971], Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton
Modular_curve
Geometric concept of a 2D space with "points at infinity" adjoined
theorem of projective geometry a reciprocity is the composition of an automorphic function of K and a homography. If the automorphism involved is the identity
Projective_plane
American mathematician (1886–1967)
reputation. In 1915 Ford published An Introduction to the Theory of Automorphic Functions as Edinburgh Mathematical Tract # 6. Returning to Harvard in 1917
Lester_R._Ford
Conjecture on zeros of the zeta function
2009-03-27 Zagier, Don (1981), "Eisenstein series and the Riemann zeta function", Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata
Riemann_hypothesis
Branch of mathematics
such as the modular group and Fuchsian group, based on work on automorphic functions in analysis. The abstract concept of group emerged slowly over the
Abstract_algebra
Theorem in number theory
transforms of weight 2 modular forms or a product of analogous automorphic L-functions. Eichler, Martin (1954), "Quaternäre quadratische Formen und die
Eichler–Shimura congruence relation
Eichler–Shimura_congruence_relation
ISSN 0012-7094, MR 0534060 Lehner, Joseph (1964), Discontinuous Groups and Automorphic Functions, Mathematical Surveys and Monographs, vol. 8, American Mathematical
Schottky_group
Branch of mathematics that studies abstract algebraic structures
of several complex variables. Automorphic forms are a generalization of modular forms to more general analytic functions, perhaps of several complex variables
Representation_theory
Conformal mappings in complex analysis
Schwarz triangle function is a single-valued automorphic function for that triangle's triangle group. More specifically, it is a modular function. Let πα, πβ
Schwarz_triangle_function
American mathematician
works in the areas of analytic number theory, automorphic forms and representation theory, L-functions, harmonic analysis, and homogeneous dynamics. Kontorovich
Alex_Kontorovich
German mathematician (1861–1930)
his work in complex analysis, especially on elliptic, modular and automorphic functions. He was one of the main collaborators of Felix Klein, with whom
Robert_Fricke
Mathematic theory
(1972), Zeta functions of simple algebras, Lect. Notes Math., vol. 260, Springer Goldfeld, Dorian; Hundley, Joseph (2011), Automorphic representations
Tate's_thesis
German mathematician (1938–2025)
ISSN 0012-9593. (online). Harder, G. (1974). "Chevalley Groups Over Function Fields and Automorphic Forms". The Annals of Mathematics. 100 (2). JSTOR: 249–306
Günter_Harder
Particular kind of exponential sum
the Riemann zeta function, primes in short intervals, primes in arithmetic progressions, the spectral theory of automorphic functions and related topics
Kloosterman_sum
French mathematician (1865–1963)
Jacques (1999) [1951]. Non-Euclidean geometry in the theory of automorphic functions. History of Mathematics. Vol. 17. Providence, R.I.: American Mathematical
Jacques_Hadamard
American mathematician
worked on automorphic functions and introduced Atkin–Lehner theory. Lehner, Joseph (1964), Discontinuous groups and automorphic functions, Mathematical
Joseph_Lehner
Interference phenomenon of waves
Theory of Diffraction", Stationary Diffraction by Wedges : Method of Automorphic Functions on Complex Characteristics, Cham: Springer International Publishing
Diffraction
Matrix equal to its transpose
JSTOR 2371774, Lemma 1, page 12 Hua, L.-K. (1944), "On the theory of automorphic functions of a matrix variable I–geometric basis", Amer. J. Math., 66 (3):
Symmetric_matrix
Theorem in complex analysis
University Press. pp. 32–38. Weiss, M. "Pólya's Shire Theorem for Automorphic Functions". Geometriae Dedicata 100, 85–92 (2003). https://doi.org/10.1023/A:1025855513977
Pólya's_shire_theorem
French mathematician, physicist and engineer (1854–1912)
field of algebraic topology, and is further credited with introducing automorphic forms. He also made important contributions to algebraic geometry, number
Henri_Poincaré
German mathematician (1862–1943)
monodromy group. 22. Uniformization of analytic relations by means of automorphic functions. 23. Further development of the methods of the calculus of variations
David_Hilbert
Riemann hypothesis. It states that the non-trivial zeros of all automorphic L-functions lie on the critical line 1 / 2 + i t {\displaystyle 1/2+it} with
Grand_Riemann_hypothesis
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 function associated to algebraic varieties
L-functions, alongside the L-functions associated to automorphic representations. Conjecturally, these two types of global L-functions are actually two descriptions
Hasse–Weil_zeta_function
Hungarian-born American mathematician (1926–2025)
ISBN 0-12-440051-5. ——; Phillips, Ralph S. (1976). Scattering Theory for Automorphic Functions. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08184-7
Peter_Lax
Soviet mathematician (1913–2009)
I.; Pyatetskii-Shapiro, I. I. (1969), Representation theory and automorphic functions, Translated from the Russian by K. A. Hirsch, Philadelphia, Pa.:
Israel_Gelfand
French mathematician (1869–1951)
not be surprising that in various areas of mathematics, such as automorphic functions and analytic number theory (apparently far removed from differential
Élie_Cartan
Japanese mathematician
especially analytic number theory, multiple trigonometric function theory, zeta functions and automorphic forms. He is currently a professor emeritus at Tokyo
Nobushige_Kurokawa
Formula in analytic number theory
was found by Kuznetsov while studying the growth of weight zero automorphic functions. Using estimates on Kloosterman sums he was able to derive estimates
Kuznetsov_trace_formula
Field extension that is not algebraic
Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Tokyo:
Transcendental_extension
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
Canadian mathematician
web of conjectures and results connecting representation theory and automorphic forms to the study of Galois groups in number theory, for which he received
Robert_Langlands
American mathematician
York City) is an American mathematician, specializing in number theory, automorphic forms, and cryptography. Hoffstein graduated with a bachelor's degree
Jeffrey_Hoffstein
Mathematical function
Mathematika. 1: 4. doi:10.1112/S0025579300000462. Bump, Daniel (1998), Automorphic Forms and Representations, Cambridge University Press, ISBN 0-521-55098-X
Dedekind_eta_function
Australian-American mathematician
Shimura, Gorō (1971). Introduction to the Arithmetic Theory of Automorphic Functions. Princeton, N.J: Princeton University Press. ISBN 978-0-691-08092-5
Frank_Calegari
Shimura, Gorō (1971). Introduction to the Arithmetic Theory of Automorphic Functions (Paperback ed.). Princeton University Press. ISBN 978-0-691-08092-5
Hecke_algebra_of_a_pair
Representation theory
ISBN 0-387-96198-4 Lax, Peter D.; Phillips, Ralph (1976), Scattering theory for automorphic functions, Annals of Mathematics Studies, vol. 87, Princeton University Press
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
German mathematician (1886–1982)
doctorate in 1910. His dissertation was titled On the theory of automorphic functions (German: Theorie der automorphen Funktionen). He began working as
Ludwig_Bieberbach
Matrices satisfying a differential equation
1002/cpa.3160210503, OSTI 4522657 archive P. Lax and R.S. Phillips, Scattering Theory for Automorphic Functions[1], (1976) Princeton University Press.
Lax_pair
Semitopological group in abstract algebra
non-archimedean places. Adelic groups provide the natural setting for automorphic forms and automorphic representations. Their basic quotients, such as G ( K ) ∖
Adelic_algebraic_group
Mathematician
Shimura, Gorō (1971). Introduction to the Arithmetic Theory of Automorphic Functions. Princeton, N.J: Princeton University Press. ISBN 978-0-691-08092-5
Yunqing_Tang
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in
Maass_wave_form
Number used for counting
a list of objects in a specific order. More precisely, a sequence is a function that assigns an object to each position in that list. The positions themselves
Natural_number
Function whose domain is the positive integers
prime-counting functions. This article provides links to functions of both classes. An example of an arithmetic function is the divisor function whose value
Arithmetic_function
Theorem in abstract algebra
In the mathematical theory of automorphic forms, the fundamental lemma relates orbital integrals on a reductive group over a local field to stable orbital
Fundamental lemma (Langlands program)
Fundamental_lemma_(Langlands_program)
Group controlling representation theory
also indicates the connection with the theory of L-functions, particularly the automorphic L-functions. The Langlands dual was introduced by Langlands (1967)
Langlands_dual_group
Special function of two variables
1962. Zagier, D. (1981). "Eisenstein series and the Riemann zeta-function". Automorphic Forms, Representation Theory and Arithmetic. Springer Berlin, Heidelberg
Real analytic Eisenstein series
Real_analytic_Eisenstein_series
Graduate-level textbooks in mathematics
Griffiths 1976-02-21 110 9780691081724 87 Scattering Theory for Automorphic Functions. Peter D. Lax, Ralph S. Phillips 1977-01-21 312 978-0691081847 88
Annals_of_Mathematics_Studies
American mathematician
referred book on scattering theory titled Scattering Theory for Automorphic Functions. Phillips received the 1997 Leroy P. Steele Prize for Lifetime Achievement
Ralph_S._Phillips
Plane algebraic curve
Serge Lang, Elliptic Functions, Addison-Wesley, 1973 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1972 OEIS
Classical_modular_curve
Theory of a class of elliptic curves
Shimura, Goro (1971). Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan. Vol. 11. Tokyo:
Complex_multiplication
German mathematician
by Weierstrass. Published in journal form in 1877, it introduced automorphic functions and Schottky groups, to be developed several years later by Henri
Friedrich_Schottky
This is a list of Lie group topics, by Wikipedia page. See Table of Lie groups for a list General linear group, special linear group SL2(R) SL2(C) Unitary
List_of_Lie_groups_topics
working in automorphic forms. He is considered one of the founders of the theory of automorphic representations and their associated L-functions, and his
Hervé_Jacquet
Complex multiplication field
Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton
CM-field
cohomology theory, again; but in general some assumption coming from automorphic representation theory seems required to get the functional equation.
Functional equation (L-function)
Functional_equation_(L-function)
Spherical triangle that can be used to tile a sphere
February 2017 Siegel, C. L. (1971), Topics in complex function theory, vol. II. Automorphic functions and abelian integrals, translated by A. Shenitzer;
Schwarz_triangle
1920s books on mathematical history by Felix Klein
among others). In the final chapter, "Group Theory and Function Theory; Automorphic Functions", Klein discusses first group theory in connection with
Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert
Vorlesungen_über_die_Entwicklung_der_Mathematik_im_19._Jahrhundert
AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, Functionary of the Interior.
Male
Egyptian
, a great 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.
Male
Celtic
, great justiciary, or functionary.
Biblical
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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
, a high Egyptian functionary.
Male
Egyptian
, an Egyptian 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.
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.
AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
Boy/Male
Tamil
Pleasing
Girl/Female
Tamil
Gunrekha | கà¯à®¨à¯à®°à¯‡à®•à®¾
Useful lines of life
Boy/Male
Arabic
Beauty
Boy/Male
Sikh
Precious like gods lamp
Boy/Male
American, British, English
Traveler
Boy/Male
Tamil
Hetaksh | ஹேதாகà¯à®·
Existence of Love
Girl/Female
English
Good Fairy
Male
Chamoru
, a stone.
Girl/Female
Indian
Love
Girl/Female
American, British, Christian, English, Hebrew
God is Gracious; God will Add; Gift of God
AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
a.
Of or pertaining to allomorphism.
adv.
In a functional manner; as regards normal or appropriate activity.
a.
Belonging or relating to life, either animal or vegetable; as, vital energies; vital functions; vital actions.
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.
v. i.
To execute or perform a function; to transact one's regular or appointed business.
pl.
of Functionary
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. t.
To assign to some function or office.
a.
Pertaining to, or connected with, a function or duty; official.
a.
Patterned after one's self.
v. i.
Alt. of Functionate
n.
Fig.: Any cavity, or hollow place, in which any function may be conceived of as operating.
prep.
Acting as a substitute; -- said of abnormal action which replaces a suppressed normal function; as, vicarious hemorrhage replacing menstruation.
a.
Pertaining to the function of an organ or part, or to the functions in general.
a.
Destitute of function, or of an appropriate organ. Darwin.
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.
n.
Automorphic characterization.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.
n.
One deputed or authorized to perform the functions of another; a substitute in office; a deputy.
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.