Functional equations

Functional equations are equations in which the unknowns are functions rather than simple variables. These equations relate the values of a function at different points in its domain and often involve operations on those functions, such as addition, multiplication, or composition. They are used to determine the forms of functions that satisfy certain conditions.

The Abel equation is a type of integral equation named after the Norwegian mathematician Niels Henrik Abel. It is typically expressed in the following form: \[ \frac{dy}{dx} = -f(y) \] where \( f(y) \) is a function of \( y \). The equation is often studied in the context of its relationship to certain integral representations, and it can also be transformed into different forms that may be more amenable to analysis.

Aequationes Mathematicae is a mathematical journal that focuses primarily on publishing research articles related to equations and mathematical analysis. Established in the mid-1970s, it covers various topics in mathematics, including differential equations, functional equations, mathematical physics, and other areas of pure and applied mathematics. The journal serves as a platform for researchers to disseminate their findings and contribute to the advancement of mathematical knowledge. It is peer-reviewed, ensuring the quality of the published work.

Böttcher's equation is a mathematical relationship used in the field of mathematics, particularly relating to complex analysis and dynamical systems. It describes the behavior of iterations of complex functions, particularly in the context of studying sequences of meromorphic functions or rational functions. In a basic sense, Böttcher's theorem provides a criterion or a condition under which a given complex dynamical system can be transformed into a simpler form, often related to the concept of normal forms.

Cauchy's functional equation is a well-known functional equation given by: \[ f(x + y) = f(x) + f(y) \] for all real numbers \(x\) and \(y\). This equation describes a function \(f\) that satisfies the property that the value of the function at the sum of two arguments is equal to the sum of the values of the function at each argument.

A **functional equation** is an equation in which the unknowns are functions, rather than simple variables. These equations establish a relationship between the values of functions at different points. Functional equations often arise in various fields of mathematics and can be used to characterize specific functions or sets of functions.