Let \(U\subset \mathbb{R}^n\) be an open subset, \(x\in U\) and \(\xi\in \mathbb{R}^n\) such that the line segment \(x+t\xi\) with \(t\in [0,1]\) is contained in \(U\). Also let \(f\colon U\to \mathbb{R}\) be a \((k+1)\) differentiable function. Then there is a \(\theta\in [0,1]\) such that
\begin{equation*} f(x+\xi)=\sum_{\lvert \alpha\rvert\le k} \frac{D^{\alpha}f(x)}{\alpha!}\xi^a+\sum_{\lvert \alpha\rvert=k+1} \frac{D^\alpha f(x+\theta \xi)}{\alpha!}\xi^\alpha. \end{equation*}
Remark
- Using the definition of derivatives we obtain for \(k\)-times differentiable functions: \begin{equation*} f(x+\xi)=\sum_{\lvert \alpha\rvert\le k} \frac{D^\alpha f(x)}{\alpha!}\xi^{\alpha} + o(\lVert \xi\rVert^k). \end{equation*}
- Some authors denote the rest with \(R_k(x,\xi)\).
- If the rest vanishes the function is analytic.