Function of several complex variables

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The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space C n {\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 the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables zi. Equivalently, they are locally uniform limits of polynomials; or solutions to the n dimensional Cauchy–Riemann equations. For one complex variable, every domain is the domain of holomorphy of some function. For several complex variables, this is not the case; there exist domains that are...

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