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Topics referred to by the same term
Function theory may refer to: Theory of functions of a real variable, the traditional name of real analysis, a branch of mathematical analysis dealing
Function_theory
Branch of mathematics studying functions of a complex variable
traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that studies complex-valued functions of one or more
Complex_analysis
Association of one output to each input
the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A function is often denoted by a letter such
Function_(mathematics)
Study of space and shapes locally given by a convergent power series
Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem
Geometric_function_theory
Field of mathematical analysis
mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation
Constructive_function_theory
Class of mathematical function
fraction ("part") of entire functions in a domain of the complex plane. In the 1930s, in group theory, the term meromorphic function (or meromorph) was used
Meromorphic_function
Fourier transform of the probability density function
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Functions in mathematics
mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle
Harmonic_function
Generalized function whose value is zero everywhere except at zero
no function has this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and
Dirac_delta_function
Mathematical framework to model epistemic uncertainty
The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty
Dempster–Shafer_theory
Study of computable functions and Turing degrees
computable function. The c.e. sets, although not decidable in general, have been studied in detail in computability theory. Beginning with the theory of computable
Computability_theory
Paradigm for the design, analysis, and scoring of tests
psychometrics, item response theory (IRT, also known as latent trait theory, strong true score theory, or modern mental test theory) is a paradigm for the design
Item_response_theory
Analytic function in mathematics
elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics
Riemann_zeta_function
Order-preserving mathematical function
was later generalized to the more abstract setting of order theory. In calculus, a function f {\displaystyle f} defined on a subset of the real numbers
Monotonic_function
One of several equivalent definitions of a computable function
If the function is total, it is also called a total recursive function (sometimes shortened to recursive function). In computability theory, it is shown
General_recursive_function
Collection of mathematical theories
generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral
Spectral_theory
Mathematical functions having established names and notations
special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely
Special_functions
Evaluation of a function on its argument
function on complete partial orders. Function application is also a continuous function in homotopy theory, and, indeed underpins the entire theory:
Function_application
Musical term
music, function (also harmonic function or tonal function) denotes the relationship of a chord or scale degree to a tonal centre. Two main theories of tonal
Function_(music)
Used to define marginal product and to distinguish allocative efficiency
fundamental elements of microeconomic production theory, see production theory basics). The production function is central to the marginalist focus of neoclassical
Production_function
Complex-differentiable (mathematical) function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood
Holomorphic_function
Generating function for quantum correlation functions
In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral
Partition function (quantum field theory)
Partition_function_(quantum_field_theory)
Number of partitions of an integer
In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because
Partition function (number theory)
Partition_function_(number_theory)
Type of mathematical functions
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space C n {\displaystyle
Function of several complex variables
Function_of_several_complex_variables
Analyzes the topology of a manifold by studying differentiable functions on that manifold
differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According
Morse_theory
Expectation value of time-ordered quantum operators
In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products
Correlation function (quantum field theory)
Correlation_function_(quantum_field_theory)
Interpretation of quantum mechanics
in de Broglie–Bohm theory is not a postulate. Rather, in this theory, the link between the probability density and the wave function has the status of
De_Broglie–Bohm_theory
Mathematical expression
use different terminology and notation for continued fractions. In number theory, the unqualified term continued fraction usually refers to simple continued
Continued_fraction
Concept in economics and decision theory
conventional foundation of choice theory in microeconomics, it is often convenient to represent preferences with a utility function. Let X be the consumption
Utility
Theoretical framework in harmonic analysis
Littlewood–Paley theory is a theoretical framework used to extend certain results about square-integrable L2 functions to Lp functions for 1 < p < ∞. It
Littlewood–Paley_theory
particular in measure theory, there are different notions of distribution function which are used in different context. Distribution functions (in the sense of
Distribution function (measure theory)
Distribution_function_(measure_theory)
foundation for a theory based on a dichotomy between "tight-knit" and "loose" designs and on "beginning," "middle," and "end" functions. The core figure
Theory_of_Formal_Function
Theory about lossy data compression
rate–distortion functions. Rate–distortion theory was created by Claude Shannon in his foundational work on information theory. In rate–distortion theory, the rate
Rate–distortion_theory
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
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
Kind of mathematical function
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves
Measurable_function
Objects extending the notion of functions
functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of
Generalized_function
Symbol representing a mathematical concept
systems particularly mathematical logic, a function symbol is a non-logical symbol which represents a function or mapping on the domain of discourse, though
Function_symbol
Finnish mathematician (1907–1996)
important field of research within function theory and later acquired significance in physics as well. His work made the theory of quasiconformal mappings a
Lars_Ahlfors
Soviet mathematician
1993) was a Soviet mathematician who made significant contributions to function theory. Boris Yakovlevich Levin was born 22 December 1906 in Odessa. In 1932
Boris_Levin
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
Branch of pure mathematics
Number theory is a branch of mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers
Number_theory
Mathematical function that preserves angles
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U {\displaystyle U} and V
Conformal_map
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
Loss_function
Correlators of field operators
In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically
Green's function (many-body theory)
Green's_function_(many-body_theory)
Harmonic functions as solutions to Laplace's equation
mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" dates from 19th-century physics when it
Potential_theory
Mathematical function that can be computed by a program
Computable functions are the basic objects of study in computability theory. Informally, a function is computable if there is an algorithm that computes
Computable_function
Set of all things that may be the input of a mathematical function
domain of the unknown function(s) sought. For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class
Domain_of_a_function
Concept in the analysis of dynamical systems
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove
Lyapunov_function
Theorem in complex analysis
analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes
Weierstrass factorization theorem
Weierstrass_factorization_theorem
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
Process by which a quantum system takes on a definitive state
mechanics. Quantum theory offers no dynamical description of the "collapse" of the wave function. Viewed as a statistical theory, no description is expected
Wave_function_collapse
Method of solution to differential equations
quantum field theory, Green's functions take the role of propagators, also referred to as two-point (correlation) functions. A Green's function, G(x,s), of
Green's_function
Mathematics of real numbers and real functions
measure theory, Lebesgue integration, and function spaces. Real analysis is also known, especially in older books, as the theory of functions of a real
Real_analysis
Mathematical models of strategic interactions
game theory include algorithmic game theory, behavioral game theory, combinatorial game theory, evolutionary game theory, and quantum game theory. In 1994
Game_theory
Mathematical study of linear operators
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The
Operator_theory
Quickly growing function
theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function
Ackermann_function
Finnish mathematician (1870–1946)
philologist Uno Lorenz Lindelöf [fi]. He founded the Finnish school of function theory, which achieved lasting international renown. He was secretary of the
Ernst_Leonard_Lindelöf
Arithmetic function related to the divisors of an integer
in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts
Divisor_function
Russian mathematician (1856–1922)
mathematician, making contributions to constructive mathematics and recursive function theory. Andrey Markov was born on 14 June 1856 in Ryazan, Russia. He attended
Andrey_Markov
Class of mathematical functions
submodular functions: Theory and enumeration algorithms", B. Goldengorin. Pseudo-Boolean function Topkis's theorem Submodular set function Superadditive
Supermodular_function
Function returning one of only two values
Boolean functions are the subject of Boolean algebra and switching theory. A Boolean function takes the form f : { 0 , 1 } k → { 0 , 1 } {\displaystyle f:\{0
Boolean_function
Extension of the domain of an analytic function (mathematics)
definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where
Analytic_continuation
Function in quantum field theory showing probability amplitudes of moving particles
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one
Propagator
Area of mathematics
mathematical field of complex analysis, Nevanlinna theory is part of the theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann
Nevanlinna_theory
Meromorphic function on the complex plane
L-function is a meromorphic function on the complex plane, and one out of several categories of mathematical objects studied in analytic number theory and
L-function
Interpretation of quantum mechanics
appearance of wave function collapse is explained by the mechanism of quantum decoherence. Decoherence approaches to interpreting quantum theory have been widely
Many-worlds_interpretation
Generalization of a positive-definite matrix
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix
Positive-definite_kernel
Characteristic property of holomorphic functions
analytic functions. Augustin-Louis Cauchy then used these equations to construct his theory of functions. Bernhard Riemann's dissertation on the theory of functions
Cauchy–Riemann_equations
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
About mathematical functions
analysis, and the invention of set theory by Georg Cantor, eventually led to the much more general modern concept of a function as a single-valued mapping from
History of the function concept
History_of_the_function_concept
Product of numbers from 1 to n
for the exponential function and other functions, and they also have applications in algebra, number theory, probability theory, and computer science
Factorial
Mathematical function characterizing set membership
indicator function in elementary number theory, the Möbius function. (See paragraph below about the use of the inverse in classical recursion theory.) Given
Indicator_function
Mathematical function with no sudden changes
theory, especially in domain theory, a related concept of continuity is Scott continuity. As a practical example, the function H(t) denoting the height of
Continuous_function
American mathematician (born 1937)
(born January 5, 1937) is an American mathematician, who works on the function theory in complex analysis. Kra studied at Polytechnic Institute of Brooklyn
Irwin_Kra
Soviet mathematician
mathematician known for contributions to real analysis and constructive function theory, in particular, for his textbooks on these subjects. His son, Garal'd
Isidor_Natanson
Theoretical framework in physics
field theory Conformal field theory Dirac's equation Form factor (quantum field theory) Green–Kubo relations Green's function (many-body theory) Group
Quantum_field_theory
Concept in stochastic analysis
builds on the integration theory of L. C. Young, the geometric algebra of Kuo-Tsai Chen, and the Lipschitz function theory of Hassler Whitney, while remaining
Rough_path
3-volume treatise on mathematics, 1910–1913
table" or "truth values" or "truth functions") and modus ponens proceeds mechanistically, by grammar alone. The theory of PM has both significant similarities
Principia_Mathematica
Mathematical theory of data types
manipulating type theories (see Logic for Computable Functions) and its own type system was heavily influenced by them. Type theory is also widely used
Type_theory
mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group G that is constant on the conjugacy
Class_function
Mathematical description of quantum state
For instance, in perturbation theory one may construct a sequence of functions approximating the true wave function. This sequence will be guaranteed
Wave_function
Concept in complex analysis
singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity)
Zeros_and_poles
British mathematician (1926–2020)
in Nevanlinna Theory. His work with Wolfgang Fuchs gave a solution to an inverse problem of the Nevanlinna theory for entire functions, predating David
Walter_Hayman
Second-order partial differential equation
function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function,
Laplace's_equation
Class of mathematical functions
superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively
Subharmonic_function
Theorem about the range of an analytic function
punctured plane by the unit disc. This function is explicitly constructed in the theory of elliptic functions. If f {\textstyle f} omits two values, then
Picard_theorem
Order-continuous function on ordinals
In set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the limits (limit suprema and limit infima)
Continuous function (ordinal theory)
Continuous_function_(ordinal_theory)
Theory of behavioral economics
theory is a theory of behavioral economics, judgment and decision making that was developed by Daniel Kahneman and Amos Tversky in 1979. The theory was
Prospect_theory
Theorem in complex analysis
poles of a meromorphic function to a contour integral of the function's logarithmic derivative. If f is a meromorphic function inside and on some closed
Argument_principle
American mathematician (1921–2010)
Bibcode:1964PhT....17a..82R. doi:10.1063/1.3051392. (1962) Function Theory in Polydiscs. (1969) Function Theory in the Unit Ball of C n {\displaystyle \mathbb {C}
Walter_Rudin
Power series derived from a discrete probability distribution
probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability
Probability generating function
Probability_generating_function
Concept in theoretical computer science
variant of the busy beaver function that was published in the proceedings of the 1964 IEEE symposium on switching circuit theory and logical design. Heiner
Busy_beaver
Method of mathematical integration
that arise in probability theory. The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general
Lebesgue_integral
Mathematical function, inverse of an exponential function
lecture, Cambridge University Press Remmert, Reinhold (1991), Theory of complex functions, New York: Springer-Verlag, ISBN 0-387-97195-5, OCLC 21118309
Logarithm
Concept of complex analysis
}{\frac {e^{itx}}{x^{2}+1}}\,dx} arises in probability theory when calculating the characteristic function of the Cauchy distribution. It resists the techniques
Residue_theorem
Conformal mappings in complex analysis
In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the
Schwarz_triangle_function
Operation on mathematical functions
two functions, f {\displaystyle f} and g {\displaystyle g} , and returns a new function f ∘ g {\displaystyle f\circ g} . When the composite function f ∘
Function_composition
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
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
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