What's the relationship between these concepts about power series expansion?

Question

A bivariate function $f:\mathbb{R}^2 \to \mathbb{R} $ is called real analytic if it can be expanded into a power series about $x-x_0$ and $y-y_0$ within a neighborhood of $(x_0,y_0)$,i.e. $$f(x,y)=\sum_{i,j=0}^{+\infty}a_{ij}(x-x_0)^i(y-y_0)^j,a_{ij}\in \mathbb{R}$$ A complex function $f:\mathbb{C}\to \mathbb{C}$ is complex analytic if and only if it can be expanded into a power series within a neighborhood of $z_0$,i.e. $$f(z)=\sum_{n=0}^{+\infty}a_n(z-z_0)^n,a_n\in \mathbb{C}$$ Moreover,we know that if a bivariate real-valued function $f\in C^{n+1}(D)$,then it has Taylor's formula $$f(x_0+h,y_0+k)=\sum_{i=0}^n \frac{(h\frac{\partial }{\partial x} +k\frac{\partial }{\partial y} )^i}{i!}f(x_0,y_0)+R_n(x_0,y_0)$$Where $R_n$ is the remainder.
My question is:What's the relationship between the first concept of real-analytic with the last two concepts?