Several complex variables

Remarks

Example: $f(z_1, z_2) = e^{z_1} \sin z_2$ is a function on $D = \C^2$. Non-example: $g(x, y) = x^2 + y^2$ on $\R^2$ is a function of two real variables, not of complex variables. It becomes a function of the complex variable $z = x + iy$ only when it is rewritten as $\abs{z}^2$ on $\C$, with $n = 1$. A map $F \colon D \to \C^m$ is the same as $m$ such functions $F_1, \dots, F_m$, its components. The case $n = 1$ is the function theory of one complex variable.