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The Euler characteristic of a point (and therefore also that of a closed or open disk) is 1. [7] Every continuous map from the closed disk to itself has at least one fixed point (we don't require the map to be bijective or even surjective); this is the case n=2 of the Brouwer fixed-point theorem. [8] The statement is false for the open disk: [9]
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.
The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of "distance" to define other "unit circles", such as the Riemannian circle; see the article on mathematical norms for additional examples.
In complex analysis, a complex domain (or simply domain) is any connected open subset of the complex plane C. For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function.
The most efficient way to pack different-sized circles together is not obvious. In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap.
The closed disk is a simple example of a surface with boundary. The boundary of the disc is a circle. The term surface used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense. A surface with empty boundary which is compact is known as a 'closed' surface.
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It follows that for a simply connected domain when ∂Ω is a simple closed curve, H 2 (∂Ω) is just the closure of the polynomials; in general it is the closure of the space of rational functions with poles lying off ∂Ω. [4] On the unit circle an L 2 function f with Fourier series expansion