In mathematics, a functional equation[1][2][irrelevant citation] is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the logarithmic functional equation
If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term functional equation is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the gamma function is a function that satisfies the functional equation and the initial value There are many functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for x real and positive (Bohr–Mollerup theorem).
^Rassias, Themistocles M. (2000). Functional Equations and Inequalities. 3300 AA Dordrecht, The Netherlands: Kluwer Academic Publishers. p. 335. ISBN 0-7923-6484-8.{{cite book}}: CS1 maint: location (link)
^ Czerwik, Stephan (2002). Functional Equations and Inequalities in Several Variables. P O Box 128, Farrer Road, Singapore 912805: World Scientific Publishing Co. p. 410. ISBN 981-02-4837-7.{{cite book}}: CS1 maint: location (link)
and 18 Related for: Functional equation information
differential equations and integral equations are functionalequations. However, a more restricted meaning is often used, where a functionalequation is an equation...
appearing in the equations A functionalequation is an equation in which the unknowns are functions rather than simple quantities Equations involving derivatives...
definition to a complex variable, proved its meromorphic continuation and functionalequation, and established a relation between its zeros and the distribution...
A functional differential equation is a differential equation with deviating argument. That is, a functional differential equation is an equation that...
first studied by Charles Babbage in 1815, and this equation is called Babbage's functionalequation. A particular solution is f(x) = (b − x)/(1 + cx) for...
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...
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions...
Hurwitz zeta function satisfies an identity which generalizes the functionalequation of the Riemann zeta function: ζ ( 1 − s , a ) = Γ ( s ) ( 2 π ) s...
Dirichlet series, it has an Euler product expansion, it satisfies a functionalequation, it has an analytic continuation to a meromorphic function on the...
A difference equation is a functionalequation that involves the finite difference operator in the same way as a differential equation involves derivatives...
{1}{1-\,\scriptstyle (-1)^{\frac {p-1}{2}}\textstyle p^{-s}}}.} The functionalequation extends the beta function to the left side of the complex plane Re(s)...
is not determined by the functionalequation, but is the limiting value of ζ(s) as s approaches zero. The functionalequation also implies that the zeta...
A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known...
→ R {\displaystyle T=T_{w}:\mathbb {R} \to \mathbb {R} } to the functionalequation T ( x ) = s ( x ) + w T ( 2 x ) . {\displaystyle T(x)=s(x)+wT(2x)...
The Abel equation, named after Niels Henrik Abel, is a type of functionalequation of the form f ( h ( x ) ) = h ( x + 1 ) {\displaystyle f(h(x))=h(x+1)}...
n, this relation is called the translation functionalequation, cf. Schröder's equation and Abel equation. On a logarithmic scale, this reduces to the...
being the Hadamard function. A more restrictive requirement is the functionalequation which interpolates the shifted factorial f ( n ) = ( n − 1 ) ! {\displaystyle...
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