Functional equations

Functional equation

In mathematics, a functional equation 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). (Wikipedia).

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Differential equation | Multiplicative function | Vector space | Logarithm | Continuous function | Mathematical analysis | Logarithmically convex function | Bohr–Mollerup theorem | Abel equation | Reflection formula | Domain of a function | Identity function | Dynamic programming | Bellman equation | Modular form | Rational number | Exponential function | Periodic function | Schröder's equation | Implicit function | Injective function | Meromorphic function | Functional differential equation | Ansatz | Linear map | List of trigonometric identities | Equation | Additive function | Gamma function | Mathematics | Natural number | Recurrence relation | Integer | Infix notation | Real number | Involution (mathematics) | Mathematical induction | Shift operator | Integral equation | Parallelogram law | Cauchy's functional equation | Functional equation (L-function) | Böttcher's equation | Functional square root | Binary operation | Hyperbolic functions | Leonhard Euler | Riemann zeta function