egységdiszknek

The egységdiszk (unit disk) is a basic object in geometry and complex analysis. It is the set of points in the Euclidean plane whose distance from the origin is at most 1. In mathematical notation, the closed unit disk is D̄ = { (x, y) ∈ R^2 : x^2 + y^2 ≤ 1 } or { z ∈ C : |z| ≤ 1 } in the complex plane. The open unit disk is D = { (x, y) ∈ R^2 : x^2 + y^2 < 1 } or { z ∈ C : |z| < 1 }.

Key properties include its center at the origin, radius 1, and boundary the unit circle S^1 = { z ∈ C : |z| = 1 }. The open disk has area π, while the boundary circle has circumference 2π. The diameter of both the open and closed unit disk is 2.

In complex analysis, the unit disk serves as a standard domain for function theory. It is common to denote it by D or U, and many results concern holomorphic functions defined on D. The automorphism group of the unit disk consists of Möbius transformations that map the disk to itself, typically of the form z ↦ e^{iθ} (z − a) / (1 − ā z) with |a| < 1. The unit disk also appears in geometry as the Poincaré disk model of hyperbolic geometry, where its interior carries a non-Euclidean metric.

Applications of the unit disk include serve as a simple test domain for convergence, mapping properties of analytic functions, and as a canonical example in various areas of analysis and geometry.