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Equation whose unknown is a function
and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates
Functional_equation
Differential equation with deviating argument
A functional differential equation is a differential equation with deviating argument. That is, a functional differential equation is an equation that
Functional differential equation
Functional_differential_equation
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
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
Mathematical formula expressing equality
appearing in the equations A functional equation is an equation in which the unknowns are functions rather than simple quantities Equations involving derivatives
Equation
of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural
Functional equation (L-function)
Functional_equation_(L-function)
Types of mappings in mathematics
equation, meaning an equation between functionals: an equation F = G {\displaystyle F=G} between functionals can be read as an 'equation to solve', with solutions
Functional_(mathematics)
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
Mathematical function, denoted exp(x) or e^x
everywhere the sum of its Maclaurin series. The exponential satisfies the functional equation exp ( x + y ) = exp ( x ) ⋅ exp ( y ) {\displaystyle \exp(x+y)=\exp(x)\cdot
Exponential_function
Computational quantum mechanical modelling method to investigate electronic structure
n one-electron Schrödinger-like equations, which are also known as Kohn–Sham equations. Although density functional theory has its roots in the Thomas–Fermi
Density_functional_theory
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 ) ζ ( 1 − s
Riemann_hypothesis
Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry) List of equations in
List_of_equations
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
Schröder's_equation
Generalization of the Riemann zeta function for algebraic number fields
{\displaystyle s=1} ; it has an Euler product expansion; and it satisfies a functional equation. Values of Dedekind zeta functions encode important arithmetic data
Dedekind_zeta_function
Type of mathematical function
L-functions satisfy a functional equation, which provides a way to analytically continue them throughout the complex plane. The functional equation relates the
Dirichlet_L-function
Equation for function that computes iterated values
The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form f ( h ( x ) ) = h ( x + 1 ) {\displaystyle f(h(x))=h(x+1)}
Abel_equation
Special mathematical function
{1}{1-\,\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)
Dirichlet_beta_function
Extension of superfactorials to the complex numbers
{\displaystyle n\to \infty } . The difference equation for the G-function, in conjunction with the functional equation for the gamma function, can be used to
Barnes_G-function
Mathematical formula
formula is an asymptotic formula for the error of the approximate functional equation of the Riemann zeta function, an approximation of the zeta function
Riemann–Siegel_formula
Schrödinger equation of a fictitious system of non-interacting particles
specifically density functional theory, the Kohn–Sham equation is the non-interacting Schrödinger equation (more clearly, Schrödinger-like equation) of a fictitious
Kohn–Sham_equations
Function that, applied twice, gives another function
first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation. A particular solution is f(x) = (b − x)/(1 + cx) for
Functional_square_root
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
Fractal curve resembling a blancmange pudding
→ R {\displaystyle T=T_{w}:\mathbb {R} \to \mathbb {R} } to the functional equation T ( x ) = s ( x ) + w T ( 2 x ) . {\displaystyle T(x)=s(x)+wT(2x)
Blancmange_curve
Functional equation characterizing associative binary operations
The associativity equation or associativity functional equation is the functional equation F ( F ( x , y ) , z ) = F ( x , F ( y , z ) ) {\displaystyle
Associativity_equation
Special function in mathematics
Hurwitz zeta function satisfies an identity which generalizes the functional equation of the Riemann zeta function: ζ ( 1 − s , a ) = Γ ( s ) ( 2 π ) s
Hurwitz_zeta_function
Type of functional equation (mathematics)
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions
Differential_equation
Böttcher's equation, named after Lucjan Böttcher, is the functional equation F ( h ( z ) ) = ( F ( z ) ) n {\displaystyle F(h(z))=(F(z))^{n}} where h
Böttcher's_equation
Concept in dynamical systems
physicist Mitchell Feigenbaum: the solution to the Feigenbaum-Cvitanović functional equation; and the scaling function that described the covers of the attractor
Feigenbaum_function
Concept in mathematics
{\displaystyle f:(0,\infty )\to \mathbb {R} } be a solution to the functional equation f ( x + 1 ) − f ( x ) = x ln x {\displaystyle f(x+1)-f(x)=x\ln
K-function
Gives a functional equation satisfied by the generating function of any rational cone
after the mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the integer-point generating function of a rational
Stanley's_reciprocity_theorem
Meromorphic function on the complex plane
absolute convergence of the Dirichlet series, analytic continuation, functional equation, Ramanujan conjecture and Euler product. This subset is today referred
L-function
Continued fraction closely related to the Rogers–Ramanujan identities
\eta (-{\tfrac {1}{\tau }})={\sqrt {-i\tau }}\,\eta (\tau )} the functional equation of the Rogers–Ramanujan continued fraction involves the golden ratio
Rogers–Ramanujan continued fraction
Rogers–Ramanujan_continued_fraction
On generating functions from counting points on algebraic varieties over finite fields
functions for smooth varieties are rational functions, satisfy a certain functional equation, and have their zeros in restricted places. The last two parts were
Weil_conjectures
Type of zeta function
a meromorphic continuation to the complex plane and satisfies a functional equation with respect to s → n − s where n is the absolute dimension of X
Arithmetic_zeta_function
Derivation of the laws of probability theory
B {\displaystyle B} given A X {\displaystyle AX} . In form of a functional equation A B ∣ X = g ( A ∣ X , B ∣ A X ) {\displaystyle AB\mid X=g(A\mid X
Cox's_theorem
Concept in probability theory
function ρ {\displaystyle \rho } ; the problem reduces to solving the functional equation ρ ( x ) ρ ( y ) = ρ ( x cos θ + y sin θ ) ρ ( x sin θ − y cos
Maxwell's_theorem
Numerical computation of special functions
It is a special case of a functional equation. It is common in mathematical literature to use the term "functional equation" for what are specifically
Reflection_formula
Implementation of the renormalization group
coupling constant. Mathematically, FRG is based on an exact functional differential equation for a scale-dependent effective action. In quantum field theory
Functional renormalization group
Functional_renormalization_group
Pattern defining an infinite sequence of numbers
and Functional Equations: Exact Solutions". at EqWorld - The World of Mathematical Equations. Polyanin, Andrei D. "Difference and Functional Equations: Methods"
Recurrence_relation
Extension of the factorial function
being the Hadamard function. A more restrictive requirement is the functional equation that interpolates the shifted factorial f ( n ) = ( n − 1 ) ! {\displaystyle
Gamma_function
Extension of the domain of an analytic function (mathematics)
is often done by first establishing some functional equation on the small domain and then using this equation to extend the domain. Examples are the Riemann
Analytic_continuation
Mathematical conjecture about zeros of L-functions
have infinitely many zeros off this line and do not satisfy the functional equation that is used to distinguish between trivial and nontrivial zeros
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Necessary condition for optimality associated with dynamic programming
A Bellman equation, named after Richard E. Bellman, is a technique in dynamic programming which breaks an optimization problem into a sequence of simpler
Bellman_equation
Hungarian mathematician
integral equations". Fenyő also made a huge number of contributions to functional equations. One of his works, "The solution of a functional equation by Laplace
István_Fenyő
branch of mathematics concerned with the study of difference (or functional) equations from the algebraic point of view. Difference algebra is analogous
Difference_algebra
Result of repeatedly applying a mathematical function
n, this relation is called the translation functional equation, cf. Schröder's equation and Abel equation. On a logarithmic scale, this reduces to the
Iterated_function
Topics referred to by the same term
a function or a partial function An alternative name for a functional equation This disambiguation page lists mathematics articles associated with the
Functional_relation
Product of numbers from 1 to n
{\displaystyle \Gamma (x-1)} are defined, the gamma function obeys the functional equation Γ ( n ) = ( n − 1 ) Γ ( n − 1 ) , {\displaystyle \Gamma (n)=(n-1)\Gamma
Factorial
Algebraic curve in mathematics
analytic continuation to the whole complex plane and satisfies a functional equation relating, for any s, L(E, s) to L(E, 2 − s). In 1999 this was shown
Elliptic_curve
Discrete analog of a derivative
A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives
Finite_difference
Collection of statistical models
OL 6477125M. "Solution of Equations by Determining the Values of the Unknown Function on a Dense Set". Lectures on Functional Equations their and Applications
Analysis_of_variance
Analytic function on the upper half-plane with a certain behavior under the modular group
function on the complex upper half-plane that roughly satisfies a functional equation with respect to the group action of the modular group and a growth
Modular_form
satisfy a number of analytic properties, including an important functional equation. The setting is in the generality of a connected quasi-split reductive
Langlands–Shahidi_method
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
Function in analytic number theory
function as a Mellin transform. Hardy gave a simple proof of the functional equation for the eta function, which is η ( − s ) = 2 1 − 2 − s − 1 1 − 2
Dirichlet_eta_function
Problem optimization method
{\displaystyle f} and g {\displaystyle \mathbf {g} } . This functional equation is known as the Bellman equation, which can be solved for an exact solution of the
Dynamic_programming
23 mathematical problems stated in 1900
in any number field. Determination of the solvability of a Diophantine equation. Quadratic forms with any algebraic numerical coefficients. Extensions
Hilbert's_problems
Mathematic theory
analysis, more precisely the Poisson summation formula, he proved the functional equation and meromorphic continuation of the zeta integral and the Hecke L-function
Tate's_thesis
Axiomatic definition of a class of L-functions
( n ε ) {\displaystyle a_{n}=O(n^{\varepsilon })} for any ε > 0; Functional equation: there is a gamma factor of the form γ ( s ) = Q s ∏ i = 1 k Γ (
Selberg_class
Functions of an angle
. One can also define the trigonometric functions using various functional equations. For example, the sine and the cosine form the unique pair of continuous
Trigonometric_functions
Simpler variant of the Riemann zeta function
zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann. Riemann's original
Riemann_xi_function
function. Dedekind introduced them in the 1880's to express the functional equation of the Dedekind eta function, in a commentary to Bernhard Riemann's
Dedekind_sum
the theory of functional equations is the following: When is it true that a function which approximately satisfies a functional equation E must be close
Cauchy–Rassias_stability
Mathematical function
equation Γ ( z + 1 ) = z Γ ( z ) . {\displaystyle \Gamma (z+1)=z\Gamma (z).\,} Taking the logarithm on both sides and using the functional equation property
Digamma_function
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
Hungarian-Canadian mathematician (1924–2020)
Aczel, was a Hungarian-Canadian mathematician, who specialized in functional equations and information theory. Aczél earned a doctorate in mathematical
János_Aczél_(mathematician)
Equation in Fourier analysis
summation can also be used to derive a variety of functional equations including the functional equation for the Riemann zeta function. One important such
Poisson_summation_formula
Polygonal curve made from right triangles
characterized axiomatically as the unique function that satisfies the functional equation f ( x + 1 ) = ( 1 + i x + 1 ) ⋅ f ( x ) , {\displaystyle f(x+1)=\left(1+{\frac
Spiral_of_Theodorus
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
{\displaystyle b(1)} and b ( 2 ) {\displaystyle b(2)} , coupled with the functional equation b ( s + 2 ) = ( s + 1 ) 2 b ( s ) , {\displaystyle b(s+2)={\frac
Lemniscate_constant
Mathematical function associated to algebraic varieties
The consequences for the functional equation were worked out by Serre and Deligne in the later 1960s; the functional equation itself has not been proved
Hasse–Weil_zeta_function
Serbian mathematician
a Serbian mathematician known for his work in differential equations, functional equations, complex analysis. He authored near 300 scientific journal
Dragoslav_Mitrinović
Inverse of a finite difference
{\displaystyle \sum _{x}f(x)=F(x)} , then F {\displaystyle F} satisfies the functional equation F ( x + 1 ) − F ( x ) {\displaystyle F(x+1)-F(x)} = f ( x ) , {\displaystyle
Indefinite_sum
Operation on mathematical functions
distribution of a function of a random variable Functional decomposition Functional square root Functional equation Higher-order function Infinite compositions
Function_composition
Field equation from quantum gravity
Wheeler–DeWitt equation for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt. The equation attempts
Wheeler–DeWitt_equation
Proposition in mathematics that is unproven
zeta-functions should be rational functions, should satisfy a form of functional equation, and should have their zeroes in restricted places. The last two
Conjecture
Greek mathematician (born 1951)
fields of Mathematical Analysis. It includes Nonlinear Functional Analysis, Functional Equations, Approximation Theory, Analysis on Manifolds, Calculus
Themistocles_M._Rassias
Size of a mathematical ball
Combining this with the values Γ(1/2) = √π and Γ(1) = 1 and the functional equation zΓ(z) = Γ(z + 1) leads to V n ( R ) = 2 π n / 2 R n n Γ ( n 2 ) =
Volume_of_an_n-ball
Infinite series summing alternating 1 and -1 terms
+ 0 + 0 + 0 + 0 + 0 + 0 + x16 + 0 + ⋯. This function satisfies a functional equation: F ( x ) = x − x 2 + x 4 − x 8 + ⋯ = x − [ ( x 2 ) − ( x 2 ) 2 +
Grandi's_series
Fundamental trigonometric functions
}\left(1-{\frac {z^{2}}{n^{2}}}\right).} sin(z) is found in the functional equation for the Gamma function, Γ ( s ) Γ ( 1 − s ) = π sin ( π s ) , {\displaystyle
Sine_and_cosine
Special function of two variables
satisfies the functional equation E ∗ ( z , s ) = E ∗ ( z , 1 − s ) , {\displaystyle E^{*}(z,s)=E^{*}(z,1-s),} analogous to the functional equation for the
Real analytic Eisenstein series
Real_analytic_Eisenstein_series
Formal power series
{F}}_{t}(z)^{x},} where 𝓕t(z) is implicitly defined by a functional equation of the form 𝓕t(z) = F(x𝓕t(z)t). Moreover, we can use matrix methods
Generating_function
Equations for correlation functions in QFT
the Green's function. They form a set of infinitely many functional differential equations, all coupled to each other, sometimes referred to as the infinite
Schwinger–Dyson_equation
Academic journal
Mathematicae is a mathematical journal. It is primarily devoted to functional equations, but also publishes papers in dynamical systems, combinatorics, and
Aequationes_Mathematicae
1957 technique for modelling problems of decision making under uncertainty
setting. In deterministic dynamic programming one usually deals with functional equations taking the following structure f t ( s t ) = max x t ∈ X t { p t
Stochastic dynamic programming
Stochastic_dynamic_programming
Functional square root of an exponential
mathematical function Schröder's equation – Equation for fixed point of functional composition Abel equation – Equation for function that computes iterated
Half-exponential_function
Simple polynomial map exhibiting chaotic behavior
map is a discrete dynamical system defined by the quadratic difference equation It is a recurrence relation and a polynomial mapping of degree 2. It is
Logistic_map
Canadian mathematician
series, and meromorphicity as well as a weak functional equation were a consequence of functional equations for Eisenstein series. This work led in turn
Robert_Langlands
the OEIS. The Dn are the unique monic polynomials satisfying the functional equation D n ( u + α u , α ) = u n + ( α u ) n , {\displaystyle D_{n}\left(u+{\frac
Dickson_polynomial
Measures the size of the ring of integers of the algebraic number field
field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K {\displaystyle K} , and the analytic
Discriminant of an algebraic number field
Discriminant_of_an_algebraic_number_field
Equation that is satisfied for all values of the variables
ISBN 978-81-7371-413-9. Efthimiou, Costas (2011). Introduction to Functional Equations (PDF). American Mathematical Society. ISBN 978-0-8218-5314-6. Archived
Identity_(mathematics)
Topics referred to by the same term
analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition Multilinear form, which generalises bilinear
Form
differential equations) Lions–Lax–Milgram theorem (partial differential equations) Lumer–Phillips theorem (semigroup theory) Marcinkiewicz theorem (functional analysis)
List_of_theorems
Algorithm for finding shortest paths
successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method. In fact, Dijkstra's
Dijkstra's_algorithm
Conjectures connecting number theory and geometry
his conjectures states that these L-functions satisfy a certain functional equation generalizing those of other known L-functions. He then goes on to
Langlands_program
Infinite series with alternating signs
research extended his work on the Basel problem and leading towards the functional equations of what are now known as the Dirichlet eta function and the Riemann
1_−_2_+_3_−_4_+_⋯
Mathematical function
Z(t)=e^{i\theta (t)}\zeta \left({\frac {1}{2}}+it\right).} It follows from the functional equation of the Riemann zeta function that the Z function is real for real
Z_function
Unsolved problem in mathematics
series. Like other L-functions, the Ramanujan L-function satisfies a functional equation: Γ ( s ) L ( s , τ ) ( 2 π ) s = Γ ( 12 − s ) L ( 12 − s , τ ) (
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Programming paradigm based on applying and composing functions
1970s, Burstall and Darlington developed the functional language NPL. NPL was based on Kleene Recursion Equations and was first introduced in their work on
Functional_programming
Basic notion of sameness in mathematics
a regular equation, called a functional equation. A functional equation involving derivatives is called a differential equation. Equations are often used
Equality_(mathematics)
Number, approximately 3.14
(}{\tfrac {n}{2}}+1{\bigr )}}}r^{n-1}.} Further, it follows from the functional equation that 2 π r = S n + 1 ( r ) V n ( r ) . {\displaystyle 2\pi r={\frac
Pi
FUNCTIONAL EQUATION
FUNCTIONAL EQUATION
Boy/Male
American, British, English
Mighty Spearman; One who Saves; The Fictional Character Jorel Father of Superman
Male
Egyptian
, a high Egyptian functionary.
Male
Egyptian
, an Egyptian functionary.
Male
Celtic
, great justiciary, or functionary.
Boy/Male
French
Fictional swordsman: (ambitious and filled with religious aspirations) from Alexander Dumas's...
Boy/Male
Australian, French
Fictional Swordsman; Ambitious and Filled with Religious Aspirations; From Alexander Dumas's Three Musketeers
Biblical
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Boy/Male
English
The fictional character Jorel father of Superman.
Boy/Male
English
The fictional character Jorel father of Superman.
Boy/Male
American, British, English
Mighty Spearman; The Fictional Character Jorel Father of Superman
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Boy/Male
English
The fictional character Jorel father of Superman.
Male
Egyptian
, Functionary of the Interior.
Boy/Male
American, Australian, British, Danish, English, Finnish, French, German, Scandinavian
Farmer; The Fictional Character Jorel Father of Superman; Earth Worker
Boy/Male
English
Modern. The fictional character Jorel father of Superman.
Male
Egyptian
, the son of the functionary Heknofre.
Boy/Male
American, Australian, British, English, French
Mighty Spearman; The Fictional Character Jorel Father of Superman
Male
Egyptian
, a great functionary.
Male
Egyptian
, an Egyptian functionary.
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.
FUNCTIONAL EQUATION
FUNCTIONAL EQUATION
Boy/Male
Arabic
Bird.
Girl/Female
Indian
One sucking her mothers milk
Girl/Female
American, Australian, Chinese, French, Greek
Victory of the People; Feminine of Nicholas; People's Victory
Girl/Female
Tamil
Lilavati | லீலாவதீ , லீலாவதீ
Goddess Durga
Girl/Female
Muslim
Abstinent
Male
English
Modern English variant spelling of French Antoine, possibly ANTWAN means "invaluable."Â
Boy/Male
Indian, Sanskrit
Celestial; Of Sacred Descent
Boy/Male
Hindu
Kingly
Girl/Female
Biblical
Beasts.
Boy/Male
British, English
Mountain Peak; Mount; Hilltop
FUNCTIONAL EQUATION
FUNCTIONAL EQUATION
FUNCTIONAL EQUATION
FUNCTIONAL EQUATION
FUNCTIONAL EQUATION
a.
Relatively small; inconsiderable; insignificant; as, a fractional part of the population.
v. t.
To supply with an organ or organs having a special function or functions.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.
pl.
of Functionary
v. i.
Alt. of Functionate
n.
The office, duties, or functions of a minister, servant, or agent; ecclesiastical, executive, or ambassadorial function or profession.
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.
a.
Pertaining to, or connected with, a function or duty; official.
a.
Pertaining to, or characterized by, fiction; fictitious; romantic.
v. i.
To execute or perform a function; to transact one's regular or appointed business.
a.
Capable of, or pertaining to, flection or inflection.
a.
Pertaining to the function of an organ or part, or to the functions in general.
n.
Paper fractional currency.
a.
Of or pertaining to fractions or a fraction; constituting a fraction; as, fractional numbers.
n.
A derived function; a function obtained from a given function by a certain algebraic process.
a.
Relating to friction; moved by friction; produced by friction; as, frictional electricity.
n.
An angle upon which the value of some function depends; -- a term used more especially in connection with elliptic functions.
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.
Fractional.
adv.
In a functional manner; as regards normal or appropriate activity.