= Several complex variables
Several complex variables studies <holomorphic functions> on open subsets of $\mathbb C^n$ and their geometry, especially when $n\geq2$. A scalar-valued function is holomorphic when it is complex differentiable in the multivariable sense; locally it has a convergent <power series> in $z_1,\ldots,z_n$. For a continuously differentiable function, the equations $\partial f/\partial\bar z_j=0$ for every $j$ give the corresponding <Cauchy-Riemann equations>.
For example, $f(z_1,z_2)=z_1z_2$ is holomorphic, while $|z_1|^2$ is not holomorphic on any open set where $z_1$ varies. A polydisc admits an iterated <Cauchy integral formula>, reconstructing interior values by integrating over one boundary circle for each coordinate.
The higher-dimensional theory is not just the one-variable theory repeated independently. <Hartogs's extension theorem> shows that isolated holes in complex dimension at least two can be removable for holomorphic functions under conditions that have no one-dimensional analogue: $1/z$ obstructs such extension in one variable. <Complex analytic spaces> and <complex analytic hypersurfaces> provide the natural geometric objects.
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