Several complex variables (nonfiction)

The theory of functions of several complex variables is the branch of mathematics dealing with complex valued functions

{\displaystyle f(z_{1},z_{2},\ldots ,z_{n})} f(z_1,z_2, \ldots, z_n) 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.