Function of several complex variables

The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space ${\displaystyle \mathbb {C} ^{n}}$, that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading.

As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables z_i. Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the n-dimensional Cauchy–Riemann equations.^[1][2][3] For one complex variable, every domain^{[note 1]}(${\displaystyle D\subset \mathbb {C} }$), is the domain of holomorphy of some function, in other words every domain has a function for which it is the domain of holomorphy.^[4][5] For several complex variables, this is not the case; there exist domains (${\displaystyle D\subset \mathbb {C} ^{n},\ n\geq 2}$) that are not the domain of holomorphy of any function, and so is not always the domain of holomorphy, so the domain of holomorphy is one of the themes in this field.^[4] Patching the local data of meromorphic functions, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem. Also, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and complex projective varieties (${\displaystyle \mathbb {CP} ^{n}}$)^[6] and has a different flavour to complex analytic geometry in ${\displaystyle \mathbb {C} ^{n}}$ or on Stein manifolds, these are much similar to study of algebraic varieties that is study of the algebraic geometry than complex analytic geometry.

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