Real analytic germs in several variables

Fix $n\ge 1$ and $a\in {\mathbb{R}}^n$. A real analytic germ at $a$ is an equivalence class of real functions agreeing on a neighborhood of $a$, represented on some polydisc $\mathrm{\mid }{x}_i-a_i\mathrm{\mid }<r_i$, $r_i>0$, by an absolutely convergent series $f(x)={\sum }_{\alpha \in {\mathbb{N}}^n}{c}_{\alpha }(x-a{)}^{\alpha }$ with real coefficients. Here $0\in \mathbb{N}$, $\alpha !={\prod }_i{\alpha }_i!$, and $c_{\alpha }=D^{\alpha }f(a)\mathrm{/}\alpha !$. A vector germ has a finite positive number of such components, with a common polydisc obtained by taking coordinatewise minima of their radii.

The multi-index convention and absolute convergence are those of Multi-indexed power series in ${\mathbb{C}}^m$ and their absolute convergence; for $n=1$ this agrees with A real-analytic function on an open subset of $\mathbb{R}$ is locally represented by a convergent real power series. The same coefficients define the complexification. Indeed choose a positive real polyradius $s<r$; absolute convergence at $a+s$ bounds $\mathrm{\mid }{c}_{\alpha }\mathrm{\mid }{s}^{\alpha }$ uniformly. An absolutely convergent multi-indexed power series is holomorphic and differentiates termwise therefore gives a holomorphic sum on the complex $s$-polydisc and the displayed derivative formula. Restriction to the real slice recovers $f$.

Gantumur, §3, Definition 12 and equation (25), printed p. 7. Real restriction and finite-vector convention are spelled out locally.