Restfunktion

Restfunktion is a term used in calculus and numerical analysis to denote the remainder that remains when a function is approximated by a polynomial, typically through a Taylor or Maclaurin expansion. If a function f is differentiable up to order n+1 at a point a, then for x near a we can write f(x) = sum_{k=0}^n f^{(k)}(a)/k! (x−a)^k + R_n(x), where R_n(x) is the restfunktion (remainder).

Common representations of the remainder include the Lagrange form and the integral form. The Lagrange form states that R_n(x) = f^{(n+1)}(ξ)/(n+1)! (x−a)^{n+1} for some ξ between a and x, provided f^{(n+1)} is continuous on the interval. The integral form expresses the remainder as R_n(x) = ∫_a^x f^{(n+1)}(t) (x−t)^n / n! dt. If the (n+1)-th derivative is bounded by M on the interval, then |R_n(x)| ≤ M |x−a|^{n+1}/(n+1)!.

In several variables, Taylor expansions around a point a yield a remainder term that is of order ||h||^{n+1} as h → 0, with analogous representations.

Example: for the exponential function, e^x = 1 + x + x^2/2! + … + x^n/n! + R_n(x), where R_n(x) = e^ξ x^{n+1}/(n+1)! for some ξ between 0 and x.

Terminology and usage: the term Restfunktion appears less commonly in modern German texts, where restterm or Restglied are often used; however, Restfunktion is still encountered in historical contexts and some mathematical discussions as the remainder viewed as a function of the expansion variable.