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Equation whose unknown is a function
a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and
Functional_equation
L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations.
Functional equation (L-function)
Functional_equation_(L-function)
Functional equation
Cauchy's functional equation is the functional equation: f ( x + y ) = f ( x ) + f ( y ) . {\displaystyle f(x+y)=f(x)+f(y).} A function f {\displaystyle
Cauchy's_functional_equation
Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry) List of equations in
List_of_equations
Formulation of classical mechanics
_{0})} into the action functional results in the Hamilton's principal function (HPF) S ( q , t ; q 0 , t 0 ) = def ∫ t 0 t L ( γ ( τ ; ⋅ ) , γ ˙ ( τ
Hamilton–Jacobi_equation
Second-order partial differential equation describing motion of mechanical system
Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The
Euler–Lagrange_equation
Differential equation with deviating argument
equation that contains a function and some of its derivatives evaluated at different argument values. Functional differential equations find use in mathematical
Functional differential equation
Functional_differential_equation
Generalized function whose value is zero everywhere except at zero
sometimes easier first to consider an equation of the form L [ u ] = h {\displaystyle L[u]=h} where h is a plane wave function, meaning that it has the form h
Dirac_delta_function
Necessary condition for optimality associated with dynamic programming
V(T(x,a))\}.} The Bellman equation is classified as a functional equation, because solving it means finding the unknown function V {\displaystyle V} , which
Bellman_equation
Description of a quantum-mechanical system
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery
Schrödinger_equation
Mathematical formula expressing equality
integral equation is a functional equation involving the antiderivatives of the unknown functions. For functions of one variable, such an equation differs
Equation
Meromorphic function on the complex plane
(-1)=-1\end{cases}}} then the extended, complete Dirichlet L-function satisfies the functional equation Λ ( χ , s ) = ϵ ( χ ) Λ ( χ ¯ , 1 − s ) . {\displaystyle
L-function
Association of one output to each input
Elementary function Functional Functional decomposition Functional predicate Functional programming Parametric equation Set function Simple function This definition
Function_(mathematics)
Mathematical function, denoted exp(x) or e^x
with the real exponential function (see § Functional equation above), the complex exponential satisfies the functional equation exp ( z + w ) = exp
Exponential_function
Generalization of the Riemann zeta function for algebraic number fields
or satisfy a functional equation. Values of Dedekind zeta functions encode important arithmetic data of K. The Dedekind zeta function is named for Richard
Dedekind_zeta_function
Computational quantum mechanical modelling method to investigate electronic structure
support of variation function δn, which is supposed to be infinitesimal. To advance toward Lagrange equation, we equate functional derivative to zero and
Density_functional_theory
Operation on mathematical functions
science) Function of random variable, distribution of a function of a random variable Functional decomposition Functional square root Functional equation Higher-order
Function_composition
Types of mappings in mathematics
In mathematics, a functional is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author)
Functional_(mathematics)
Equation for fixed point of functional composition
Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function h, find the function Ψ such that
Schröder's_equation
Mathematical concept
GL(n). The resulting Rankin-Selberg L-functions satisfy a number of analytic properties, their functional equation being first proved via the Langlands–Shahidi
Automorphic_L-function
Analytic function in mathematics
definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution
Riemann_zeta_function
Function that, applied twice, gives another function
mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition
Functional_square_root
Equations of motion for viscous fluids
calculus). The first equation is a pressureless governing equation for the velocity, while the second equation for the pressure is a functional of the velocity
Navier–Stokes_equations
Concept in calculus of variations
functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions)
Functional_derivative
Formulation of classical mechanics using momenta
Schrödinger equation. The value of the Hamiltonian H {\displaystyle {\mathcal {H}}} is the total energy of the system if and only if the energy function E L {\displaystyle
Hamiltonian_mechanics
Result of repeatedly applying a mathematical function
equation Functional square root Abel equation Böttcher's equation Infinite compositions of analytic functions Flow (mathematics) Tetration Functional
Iterated_function
Mathematic theory
summation formula, he proved the functional equation and meromorphic continuation of the zeta integral and the Hecke L-function. He also located the poles of
Tate's_thesis
Type of mathematical function
{\displaystyle s=1} . For generalizations, see the article on functional equations of L-functions. Let χ {\displaystyle \chi } be a primitive character modulo
Dirichlet_L-function
Type of Dirichlet series associated to number field extensions
number. This functional equation generalizes equations for Hecke L-functions and Dedekind zeta functions, especially archetypical equation for Riemann
Artin_L-function
Mathematical function associated to algebraic varieties
the Hasse–Weil zeta function should extend to a meromorphic function for all complex s, and should satisfy a functional equation similar to that of the
Hasse–Weil_zeta_function
Type of functional equation (mathematics)
a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent
Differential_equation
theorem Elliott–Halberstam conjecture Functional equation (L-function) Chebotarev's density theorem Local zeta function Weil conjectures Modular form modular
List_of_number_theory_topics
Partial differential equation
the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the position or
Fokker–Planck_equation
Mathematical relation consisting of a multi-variable function equal to zero
an implicit equation is a relation of the form R ( x 1 , … , x n ) = 0 , {\displaystyle R(x_{1},\dots ,x_{n})=0,} where R is a function of several variables
Implicit_function
Technique to solve partial differential equations
following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ
Physics-informed neural networks
Physics-informed_neural_networks
Eisenstein series. The resulting L-functions satisfy a number of analytic properties, including an important functional equation. The setting is in the generality
Langlands–Shahidi_method
Type of differential equation
differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought
Partial_differential_equation
Field equation from quantum gravity
the wave function is Ψ [ γ , ω ] {\displaystyle \Psi [\gamma ,\omega ]} . The Wheeler–DeWitt equation is a functional differential equation. It is ill-defined
Wheeler–DeWitt_equation
Auxiliary functions used to probe equations, distributions, and weak formulations
Test functions are auxiliary functions used in mathematical analysis to probe other functions, distributions, differential equations, or variational identities
Test_function
Mathematical function
Taking the logarithm on both sides and using the functional equation property of the log-gamma function gives: log Γ ( z + 1 ) = log ( z ) + log Γ
Digamma_function
analytic continuation, and the automorphy of the theta function to prove the functional equation. Erich Hecke, and later Hans Maass, applied the same Mellin
Rankin–Selberg_method
Formulation of quantum mechanics
this equation by another functional S ^ = S − i ln M . {\displaystyle {\hat {\mathcal {S}}}={\mathcal {S}}-i\ln M.} If we expand this equation as a
Path-integral_formulation
Stochastic differential equation
In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination
Langevin_equation
Simpler variant of the Riemann zeta function
Riemann xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named
Riemann_xi_function
Special mathematical functions defined on the surface of a sphere
harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific
Spherical_harmonics
Programming paradigm based on applying and composing functions
computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative
Functional_programming
Partial differential equations whose solutions are instantons
Euler–Lagrange equations of the Yang–Mills action functional. They have also found significant use in mathematics. Solutions of the equations are called Yang–Mills
Yang–Mills_equations
Differential calculus on function spaces
involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus
Calculus_of_variations
Study of classical optics using Fourier transforms
interested reader may investigate other functional linear operators (so for different equations than the Helmholtz equation) which give rise to different kinds
Fourier_optics
Mathematical function having a characteristic S-shaped curve or sigmoid curve
of partial differential equation (PDE) M21: Solutions of functional differential equation (FDE) M22: Sum of a sigmoid function and some derivatives M23:
Sigmoid_function
it is the functional derivative (a function S → R {\displaystyle S\rightarrow \mathbb {R} } ) the Poisson matrix L ( x ) {\displaystyle L(x)} is an antisymmetric
GENERIC_formalism
Extension of the factorial function
function is known as a pseudogamma function, the most famous being the Hadamard function. A more restrictive requirement is the functional equation that
Gamma_function
Artin as an expression appearing in the functional equation of an Artin L-function. Suppose that L {\displaystyle L} is a finite Galois extension of the
Artin_conductor
Half maximal inhibitory concentration
competitive antagonist affinity from functional inhibition curves using the Gaddum, Schild and Cheng–Prusoff equations". British Journal of Pharmacology
IC50
Functions in mathematics
"harmonic function" originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type
Harmonic_function
Physical quantity of dimension energy × time
principle results in the equations of motion in Lagrangian mechanics. In addition to the action functional, there is another functional called the abbreviated
Action_(physics)
Finding linear approximation of function at given point
Stability derivatives Linearization theorem Taylor approximation Functional equation (L-function) Quasilinearization The linearization problem in complex dimension
Linearization
the generating functional for the connected Green's function. As an example, the two particle connected Green's function reads: G i j k l c o n n = − ⟨
Luttinger–Ward_functional
Diagram showing the proportion of a receptor bound to a ligand
pharmacology, the Hill equation refers to two closely related equations that reflect the binding of ligands to macromolecules, as a function of the ligand concentration
Hill_equation_(biochemistry)
Differential equations involving stochastic processes
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution
Stochastic differential equation
Stochastic_differential_equation
a sought function. The logarithm of this functional equation amounts to Schröder's equation. Solution of functional equation is a function in implicit
Böttcher's_equation
Theorem In probability theory and statistics
Campbell–Hardy theorem is either a particular equation or set of results relating to the expectation of a function summed over a point process to an integral
Campbell's theorem (probability)
Campbell's_theorem_(probability)
Mathematical model of waves on a shallow water surface
In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow
Korteweg–De_Vries_equation
Equation describing a state of matter under a given set of conditions
In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given
Equation_of_state
Integral transform useful in probability theory, physics, and engineering
functional equation of the Riemann zeta function, and his method is still used to relate the modular transformation law of the Jacobi theta function,
Laplace_transform
value of the function. A primitive can be defined by an equation or by a "black box" procedure converting point coordinates into the function value. Solids
Function_representation
Mathematical description of quantum state
string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle
Wave_function
Nowhere analytic, infinitely differentiable function
f(1-x)=1-f(x)} for 0 ≤ x ≤ 1 {\displaystyle 0\leq x\leq 1} , and the functional differential equation f ′ ( x ) = 2 f ( 2 x ) {\displaystyle f'(x)=2f(2x)} for
Fabius_function
Class of ordinary differential equations
L mapping a function u to another function Lu, and it can be studied in the context of functional analysis. In fact, equation (1) can be written as L
Sturm–Liouville_theory
Branch of mathematical analysis
the functional square root for the differentiation operator, that is, an expression for some linear operator that, when applied twice to any function, will
Fractional_calculus
Branch of functional analysis
means the kind of function of an operator which is allowed. The Borel functional calculus is more general than the continuous functional calculus, and its
Borel_functional_calculus
Statistical function that defines the quantiles of a probability distribution
inverse c.d.f., the quantile is a (potentially) set valued functional of a distribution function F, given by the interval Q ( p ) = [ sup { x : F ( x ) <
Quantile_function
Formal power series
expansion of the last generating function. One often encounters generating functions specified by a functional equation, instead of an explicit specification
Generating_function
Equations with an unknown function under an integral sign
integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be
Integral_equation
Type of character in number theory
Hecke showed these L-functions satisfy a functional equation relating the values of the L-function of a character and the L-function of its complex conjugate
Hecke_character
Equations that describe the behavior of a physical system
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically
Equations_of_motion
Method for load calculation in construction
{\displaystyle q(x)} . The Euler–Lagrange equation is used to determine the function that minimizes the functional S {\displaystyle S} . For a dynamic Euler–Bernoulli
Euler–Bernoulli_beam_theory
the entire complex plane and satisfies a functional equation relating the L-function L(s, M) of a motive M to L(1 − s, M∨), where M∨ is the dual of the
Motivic_L-function
Numerical method
An adjoint state equation is introduced, including a new unknown variable. The adjoint method formulates the gradient of a function towards its parameters
Adjoint_state_method
Differential operator in mathematics
of that density distribution. Solutions of Laplace's equation Δf = 0 are called harmonic functions and represent the possible gravitational potentials
Laplace_operator
The stability problem of functional equations originated from a question of Stanisław Ulam, posed in 1940, concerning the stability of group homomorphisms
Hyers–Ulam–Rassias_stability
Economic formula of productivity
econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the relationship
Cobb–Douglas production function
Cobb–Douglas_production_function
Formulation of classical mechanics
= ∫ t 1 t 2 L d t , {\displaystyle S=\int _{t_{1}}^{t_{2}}L\,\mathrm {d} t,} which is a functional; it takes in the Lagrangian function for all times
Lagrangian_mechanics
Special function in mathematics
{\displaystyle 1} . The Hurwitz zeta function satisfies an identity which generalizes the functional equation of the Riemann zeta function: ζ ( 1 − s , a ) = Γ ( s
Hurwitz_zeta_function
Special mathematical function
\scriptstyle (-1)^{\frac {p-1}{2}}\textstyle p^{-s}}}.} The functional equation extends the beta function to the left side of the complex plane Re(s) ≤ 0. It
Dirichlet_beta_function
Mathematical transform that expresses a function of time as a function of frequency
transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can be formally defined as
Fourier_transform
Constants of the mathematical zeta function
zeta function is difficult, often certain ratios can be found by inserting particular values of the gamma function into the functional equation ζ ( s
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
Mathematical formula
for the error of the approximate functional equation of the Riemann zeta function, an approximation of the zeta function by a sum of two finite Dirichlet
Riemann–Siegel_formula
Type of differential equation
mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given
Delay_differential_equation
Generalization of the inverse function theorem
prove local existence for non-linear partial differential equations in spaces of smooth functions. It is particularly useful when the inverse to the derivative
Nash–Moser_theorem
Conjecture on zeros of the zeta function
0<\operatorname {Re} (s)<1} this extension of the zeta function satisfies the functional equation ζ ( s ) = 2 s π s − 1 sin ( π s 2 ) Γ ( 1 − s )
Riemann_hypothesis
Axiomatic definition of a class of L-functions
violates the Riemann hypothesis. Without this functional equation we would have Dirichlet L-functions for any imprimitive character. If χ {\textstyle
Selberg_class
Solution with prescribed norm
considered Schrödinger equations with a general nonlinear term. In the case the functional is not bounded below, i.e., the L 2 {\displaystyle L^{2}} supercritical
Normalized solution (mathematics)
Normalized_solution_(mathematics)
Asymmetric sigmoid function
of individuals living at a given age as a function of age. Earlier work on the construction of functional models of mortality was done by the French
Gompertz_function
Equations for correlation functions in QFT
Schwinger–Dyson equations (SDEs) or Dyson–Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between correlation functions in
Schwinger–Dyson_equation
Extension of the factorial function
function, Hadamard's gamma function H(x) is an entire function, i.e., it is defined and analytic at all complex numbers. It satisfies the functional equation
Hadamard's_gamma_function
Manifold of functions in the calculus of variations
generally, given a suitable functional J, the associated Nehari manifold is defined as the set of functions u in an appropriate function space for which ⟨ J ′
Nehari_manifold
Linear differential equation
linear functional: J ( u ) = ∫ Ω g u d V . {\displaystyle J(u)=\int _{\Omega }gu\ dV.} Derive the weak form by multiplying the primal equation with a
Adjoint_equation
Technique for solving differential equations
biharmonic equation above). Consider an initial boundary value problem for a function u ( x , t ) {\displaystyle u(x,t)} on D = { ( x , t ) : x ∈ [ 0 , l ] ,
Separation_of_variables
In mathematics, a solution to a modified form of the confluent hypergeometric equation
mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1903)
Whittaker_function
FUNCTIONAL EQUATION-L-FUNCTION
FUNCTIONAL EQUATION-L-FUNCTION
Male
Irish
Irish Gaelic form of Greek MichaÄ“l, MÃCHEÃL means "who is like God?"
Male
Celtic
, great justiciary, or functionary.
Male
French
French form of Hebrew Rephael, RAPHAËL means "healed of God" or "whom God has healed."
Male
Egyptian
, an Egyptian functionary.
Male
Swedish
Swedish form of Greek Paulos, PÃ…L means "small."
Male
Egyptian
, Functionary of the Interior.
Male
French
French form of Greek Ioel (Hebrew Yowel), JOËL means "Jehovah is God" or "to whom Jehovah is God."
Male
French
Masculine form of French Gaëlle, GAËL means "holy and generous."
Boy/Male
Irish
Rooster.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Dutch
, God's judge.
Male
Egyptian
, a high Egyptian functionary.
Male
Egyptian
, an Egyptian functionary.
Male
Irish
Irish form of Greek Paulos, PÓL means "small."
Male
French
French name derived from Latin natalis dies, NOËL means "day of birth."
Male
Hungarian
Hungarian form of Roman Latin Cornelius, KORNÉL means "of a horn."
Male
Egyptian
, a great functionary.
Male
Hungarian
Hungarian form of Greek Paulos, PÃL means "small."
Male
Norwegian
Norwegian variant form of Scandinavian Njal, NJÃ…L means "champion."
Male
Scottish
Scottish form of Latin Paulus, PÀL means "small."
FUNCTIONAL EQUATION-L-FUNCTION
FUNCTIONAL EQUATION-L-FUNCTION
Boy/Male
Swedish German
Earnest.
Boy/Male
Hindu
Vi-without, Shank-fear/hesitation/doubt, Vishank = one who knows no fear, No hesitation, No doubts
Girl/Female
Finnish
consecrated to God.
Boy/Male
Arabic, French, Hindu, Indian, Muslim
Rebellious; Ray of Light
Boy/Male
Tamil
Healthy, Vanity, Breath, Breathing
Boy/Male
Muslim
The bestower of honour
Boy/Male
Muslim
The one who believes in oneness of Allah almighty
Boy/Male
American, Australian, British, Christian, English, French, German
Peaceful; God's Peace
Girl/Female
Celtic
Divine one.
Boy/Male
French
Pasture of oats.
FUNCTIONAL EQUATION-L-FUNCTION
FUNCTIONAL EQUATION-L-FUNCTION
FUNCTIONAL EQUATION-L-FUNCTION
FUNCTIONAL EQUATION-L-FUNCTION
FUNCTIONAL EQUATION-L-FUNCTION
a.
Relating to friction; moved by friction; produced by friction; as, frictional electricity.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.
n.
The process of separating a fusible substance from one less fusible, by means of a degree of heat sufficient to melt the one and not the other, as an alloy of copper and lead; liquation.
n.
The act or process of educating; the result of educating, as determined by the knowledge skill, or discipline of character, acquired; also, the act or process of training by a prescribed or customary course of study or discipline; as, an education for the bar or the pulpit; he has finished his education.
v. i.
To execute or perform a function; to transact one's regular or appointed business.
n.
See L.
adv.
In a functional manner; as regards normal or appropriate activity.
a.
Pertaining to, or connected with, a function or duty; official.
n.
An extension at right angles to the length of a main building, giving to the ground plan a form resembling the letter L; sometimes less properly applied to a narrower, or lower, extension in the direction of the length of the main building; a wing.
n.
An expression of the condition of equality between two algebraic quantities or sets of quantities, the sign = being placed between them; as, a binomial equation; a quadratic equation; an algebraic equation; a transcendental equation; an exponential equation; a logarithmic equation; a differential equation, etc.
n.
The process of separating, by heat, an easily fusible metal from one less fusible; eliquation.
a.
Relatively small; inconsiderable; insignificant; as, a fractional part of the population.
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.
A biquadratic equation.
n.
An identical equation.
v. i.
Alt. of Functionate
a.
Of or pertaining to fractions or a fraction; constituting a fraction; as, fractional numbers.
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
a.
Pertaining to the function of an organ or part, or to the functions in general.