Holomorphic map

Remarks

Equivalently, $F$ is differentiable at each point $a \in U$ in the real sense, and its derivative $DF(a) \colon \C^n \to \C^m$ is $\C$-linear. This derivative is given by the complex Jacobian matrix $\left( \frac{\partial F_k}{\partial z_j}(a) \right)_{k, j}$, an $m \times n$ matrix. Example: $F(z_1, z_2) = (z_1 z_2, \ z_1 - z_2)$ is holomorphic from $\C^2$ to $\C^2$, since both components are polynomials. Non-example: $G(z_1, z_2) = (z_1, \ \bar z_2)$ is not holomorphic, since its second component fails the Cauchy–Riemann equation $\partial G_2 / \partial \bar z_2 = 0$. Compositions of holomorphic maps are holomorphic, by the chain rule.