: Astronomy: functional equation

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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 log⁡ (x y) = log (x) + log⁡ (y). 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 f(x + 1) = xf(x) and the initial value f(1) = 1. 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, The Free Encyclopedia, https://en.wikipedia.org/wiki/Functional_equation)

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