Several complex variables is a branch of mathematics that extends complex analysis, which traditionally deals with functions of a single complex variable, to functions that take several complex variables as input. It studies the properties and applications of functions of multiple complex variables, examining aspects such as holomorphicity (the complex analogue of differentiability), singularities, and complex manifolds.
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Several complex variables studies holomorphic functions on open subsets of and their geometry, especially when . A scalar-valued function is holomorphic when it is complex differentiable in the multivariable sense; locally it has a convergent power series in . For a continuously differentiable function, the equations for every give the corresponding Cauchy-Riemann equations.
For example, is holomorphic, while is not holomorphic on any open set where 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: obstructs such extension in one variable. Complex analytic spaces and complex analytic hypersurfaces provide the natural geometric objects.