Definition (3.19)

Let $\mathcal{U}\subset \mathbb{R}^{n}$ be an Open set and and let $f:\mathcal{U}\to \mathbb{R}^{m}$ be smooth. The Taylor series for $f$ about $\boldsymbol x_{0}$ is the formal power series $$\sum_{r=0}^{+\infty} \frac{1}{r!}\boldsymbol D^{r}f(\boldsymbol x_{0})\cdot \underbrace{(\boldsymbol v, \dots, \boldsymbol v)}_{r\text{ copies}},\quad\boldsymbol v\in \mathbb{R}^{n}$$

Definition (3.19b)

Let $\mathcal{U}\subset \mathbb{R}^{n}$ be an Open set and and let $f:\mathcal{U}\to \mathbb{R}^{m}$ be smooth. The function $f$ is real-analytic if, for every $\boldsymbol x_{0}\in \mathcal{U}$, there exists $\rho>0$ such that the Taylor Series converges to $$f(\boldsymbol x_{0}+\boldsymbol v)\text{ provided that }\lVert \boldsymbol v \rVert _{\mathbb{R}^{n}}<\rho.$$A real-analytic map is said to be of class $\boldsymbol C^{\boldsymbol \omega}$.