Logarithmusreihe

Logarithmusreihe is a mathematical concept that involves a sequence of logarithmic functions. It is a way to extend the domain of a logarithmic function beyond the positive real numbers by considering a sequence of functions, each of which is obtained by composing the original logarithmic function with a shift in its argument. This can be seen as a way to deal with the fact that a logarithmic function is not defined at the value 0.

In mathematical terms, given a real number x, a logarithmic function f(x) is a function such that f(a) = f(b) if and only if a = eb for some real number b. The logarithmusreihe can be defined as a sequence of functions f_n(x), where for each n, f_n(x) = f(f_(n-1)(x)). This results in a sequence of functions that converge to a function f(x) as n approaches infinity.

The concept of logarithmusreihe is closely related to the theory of continued fractions. In fact, the logarithmusreihe can be viewed as a special case of a continued fraction where the coefficients are all equal to 1. This connection to continued fractions has led to the development of more general theories, including the study of arithmetic like to be Dedekind-chains and orders, which involve algebraic objects, such as groups and particular defined monoids which allow to construct rationals and rational-like numbers.