Functional equation

In Mathematical, a functional equation is a equation whose unknown factors are functions. Many properties of functions can be given while studying which equations they satisfy. Usually, the term functional equation is reserved for the equations which one cannot not directly bring back to an algebraic equation, generally because the sought function has as arguments in the equation, the variable, but of the functions (given) of the variable itself.

Examples

the functional equation

\ zeta (S) = 2^s \ pi^ {s-1} \ sin \ left (\ frac {\ pi S} {2} \ right) \ Gamma (1-s) \ zeta (1-s)

is satisfied by the Fonction Zeta with Riemann. Letter Γ indicates the Fonction Gamma of Euler.

the functional equation

$x \ Gamma (X) = \ Gamma (x+1)$

is satisfied by the Fonction Gamma with Euler.

the functional equation

$f \ left ({az+b \ over cz+d} \ right) = (cz+d)^k F (Z)$

where has , B , C and D are whole natural checking AD − bc = 1, finds in the definition of the modular concept of Forme.

Of other examples of functional equations:

F ( X + there ) = F ( X ) F ( there ), satisfied by the exponential functions

F ( xy ) = F ( X ) + F ( there ), satisfied by the functions logarithms

F ( X + there ) = F ( X ) + F ( there ) (equation of Cauchy)

F ( az ) = aF ( Z ) (1 − F ( Z )) (equation of Poincaré)

G ( X ) = λ⁻¹ G ( G (λ Z )) (theory of chaos)

F (( X + there ) /2) = ( F ( X ) + F ( there ))/2 (Jensen)

G ( X + there ) + G ( X − there ) = 2 G ( X ) G ( there ) (of Alembert)

F ( H ( X )) = cf ( X ) (Schröder)

F ( H ( X )) = F ( X ) + 1 (Abel).

a simple form of functional equation is the relation of recurrence, which includes/understands an unknown function definite on the whole of the entireties and the operator of translation.

an example of relation of recurrence:

$has (N) = 3a (n-1) + 4a (N2)$

the commutative and associative laws are functional equations. When the associative law is expressed in its usual form, one represents a binary operation by a symbol between the two variables, as follows:

( has * B ) * C = has * ( B * C ),

But if one writes F ( has , B ) instead of has * B , then the associative law resembles more than one understands conventionally by " equation fonctionnelle":

F ( F ( has , B ), C ) = F ( has , F ( B , C )).

A common point with all these examples is that in each of the cases, two or several functions (sometimes multiplication by a constant, sometimes addition of two variables, sometimes the function identity) are substituted for the unknown factor.

When it is question of finding all the solutions, it happens that certain analytical conditions are required; for example, in the case of the equation of Cauchy, the continuous solutions are the reasonable solutions whereas the other solutions are accessible with more difficulty. The theorem of Bohr - Mollerup is another known example.

See too

Differential equation

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