Expansionpunkt

Expansionpunkt (Expansion point) is a term used in mathematics and related fields to denote the point around which a function or dataset is expanded into a series. It acts as the center of expansion, shaping the local approximation and the form of the resulting series. The choice of expansion point influences convergence properties and the accuracy of the approximation in the neighborhood of that point.

In calculus, the Taylor series of a function f differentiable near a is expressed as f(x) = sum_{n=0}^∞ f^(n)(a)/n! (x - a)^n. The point a is the expansionpunkt. If a = 0, the series is called the Maclaurin series. The series converges within a radius R, such that |x - a| < R, determined by the function’s analytic properties. If a is a singular point, one uses a Laurent expansion with negative powers.

Choosing an expansionpunkt close to the region of interest improves approximation accuracy with a finite truncation. In numerical analysis and applied modeling, one may use piecewise expansions about different points to cover larger domains. In physics and engineering, linearization around a particular operating point (the expansionpunkt) yields a local, first-order model valid for small deviations.

Beyond real functions, the concept extends to complex analysis, where Taylor and Laurent expansions are taken about a complex point. The term expansionpunkt is used in German-language texts and is equivalent to center of expansion or expansion center in English. It remains a standard concept for local approximation and perturbative methods.