Analytic functions in higher dimensions? We say that an analytic function is a function that is locally given by a convergent power series. I am wondering how the analytic function could be in higher dimensions? In other words, how to write the power series for a function $f:\mathbb{R}^n\to \mathbb{R}$? 
If it is given by the partial derivatives of $f$?
 A: It's quite ugly in general form, but the multivariable series is given by
$$ T(x_{1},\ldots ,x_{j})=\sum _{n_{1}=0}^{\infty }\cdots \sum _{n_{j}=0}^{\infty }{\frac {(x_{1}-a_{1})^{n_{1}}\cdots (x_{j}-a_{j})^{n_{j}}}{n_{1}!\cdots n_{j}!}}\,\left({\frac {\partial ^{n_{1}+\cdots +n_{j}}f}{\partial x_{1}^{n_{1}} \cdots \partial x_{j}^{n_{j}}}}\right)(a_{1},\ldots ,a_{j})$$
A: A function $f : \Omega (\subseteq \mathbb C^n) \to \mathbb C$ of several complex variables is called holomorphic if it is holomorphic in each variable separately.
This is equivalent to for each $z_0 \in \Omega$ there be an $r > 0$ such that $\overline{D^n}(z_0, r) \subseteq \Omega$ and
$$f(z) = \sum_\alpha a_\alpha(z-z_0)^\alpha$$
where the series converges absolutely and uniformly. Here,
$$D^n(z_0, r) = \{ z = (z_j)_{j=1}^{n} \in \mathbb C^n : |z_j - z_{0j}| < r; j = 1, \ldots, n \}$$
and $\alpha$ is a multi-index. The bar over $D^n$ denotes closure.
Source: Function Theory of Several Complex Variables, 2nd Ed., Steven G. Krantz
