Holomorphic function of several variables

Remarks

The map $L$ is the Fréchet derivative of $f$ at $a$, and it has the form $L(h) = \sum_{j=1}^n \frac{\partial f}{\partial z_j}(a) \, h_j$. Example: $f(z_1, z_2) = z_1 z_2$ is holomorphic on $\C^2$, since $f(a + h) - f(a) = a_2 h_1 + a_1 h_2 + h_1 h_2$ and $\abs{h_1 h_2} \le \abs{h}^2$. Non-example: $g(z_1, z_2) = \bar z_1$ is real differentiable with derivative $h \mapsto \bar h_1$, but this map is not $\C$-linear, since it sends $i h$ to $-i \bar h_1$. Equivalent forms: $f$ is holomorphic on $D$ if and only if $f$ is continuous and holomorphic in each variable separately. It is also equivalent that $f$ is, near each point $a$, the sum of a convergent power series in several variables centred at $a$. By Hartogs's theorem on separate holomorphicity, the continuity condition in the first equivalent form may even be dropped. The set of holomorphic functions on $D$ is written $\mathcal{O}(D)$.