The following will work extensively with holomorphic functions on complex spaces. While these objects are pretty much omnipresent, but some other definitions might be in order. A polydisc is a product defined on n-dimensional complex coordinates and an n-dimensional vector r.
Holomorphic complex functions gain satisfy the Cauchy-Riemann equations, meaning that they also have a Cauchy integral representation. In fact, these properties are equivalent for continuous functions on open sets. A pair of holomorphic functions in a region, that agree in a neighborhood of some point within said region will agree throughout the entire region. If F is holomorphic on a region O containing the union of K₁ = {(z₁, z₂): |z₁| ≤ a, |z₂| = b₁}, K₂ = {(z₁, z₂): |z₁| = a, b₂ ≤ |z₂| ≤ b₁}, then it extends analytically to an open set containing the product set {(z₁, z₂): |z₁| ≤ a, b₂ ≤ |z₂| ≤ b₁}. This also means that if F is holomorphic in Ω = {z \in ℂⁿ, ρ < |z| < 1} for fixed ρ between 0 and 1, it can be analytically continued into the a complex n-Ball with radius 1. For n > 1, the function can't have an isolated singularity or isolated zero, and holomorphic functions inside the unit ball can't necessarily be extended outside it. The generalized study of these analytical continuations requires some use of inhomogeneous Cauchy-Riemann equations.
The case of n = 1 is easy and immediately solvable, the complexity of the solution scales with dimension. If f is continuous with compact support on ℂ, then the solution is given by this one-dimensional solution, and satisfies the one-dimensional Cauchy-Riemann equations. If f is in the class Cᵏ, k > 0, then its solution satisfies the one-dimensional Cauchy-Riemann equation as well. Functions in C¹ with compact support are automatically solutions of the 1D Cauchy-Riemann equation. If n > 1, then f must be functions of class Cᵏ of compact support, satisfying the CR consistency condition
and a function u exists satisfying the inhomogeneous CR equations. This yields a general form of Hartog's principle for ℂⁿ, n ≥ 2, and K a compact subset of Ω with connected Ω - K. Any function analytic in Ω - K has an analytic continuation in Ω. A defining function in ℝᵈ is a real valued function with
if ρ is of class Cᵏ, and |∇ρ(x)| > 0, then Ω is considered to be of class Cᵏ as well. The boundary then is a hypersurface of class Cᵏ. A local hypersurface M of class Cᵏ is given by a real Cᵏ function on a ball in ℝᵈ, so that the points in the ball evaluate said function to zero.
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