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The traditional usage also applies when one talks about a functional equation, meaning an equation between functionals: an equation = between functionals can be read as an 'equation to solve', with solutions being themselves functions. In such equations there may be several sets of variable unknowns, like when it is said that an additive map is ...
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 ).
For functions of one variable, such an equation differs from integral and differential equations through a similar change of variable. A functional differential equation of delay differential equation is a function equation involving derivatives of the unknown functions, evaluated at multiple points, such as ′ = ()
Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction, multiplication, and division (without the need of taking limits).
The Bellman equation is classified as a functional equation, because solving it means finding the unknown function , which is the value function. Recall that the value function describes the best possible value of the objective, as a function of the state x {\displaystyle x} .
Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation). When there is no algebraic expression for the roots, and when such an algebraic expression exists but is too complicated to be useful, the unique way of solving it is to compute numerical approximations of the ...