Several complex variable
The theory of functions of several complex variables is the branch of mathematics dealing with complex valued functions
on the space Cn of n-tuples of complex numbers. As in complex analysis, which is the case n = 1 but of a distinct character, these are not just any functions: they are supposed to be holomorphic or complex analytic, so that locally speaking they are power series in the variables zi.
Equivalently, as it turns out, they are locally uniform limits of polynomials; or local solutions to the n-dimensional Cauchy–Riemann equations.
Many examples of such functions were familiar in nineteenth-century mathematics: abelian functions, theta functions, and some hypergeometric series. Naturally also any function of one variable that depends on some complex parameter is a candidate. The theory, however, for many years didn't become a full-fledged area in mathematical analysis, since its characteristic phenomena weren't uncovered. The Weierstrass preparation theorem would now be classed as commutative algebra; it did justify the local picture, ramification, that addresses the generalisation of the branch points of Riemann surface theory.
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