Table of Contents
What is a punctured disc?
A punctured disk can be described as an open annulus with center x0 and an inner radius of zero. The region is defined by two inequalities. 0 < |z – z0| < R, where R = the punctured disk’s radius. The radius can be any positive number; when R is infinity, the region is usually called a punctured plane (Sarason, 2007).
What is punctured circle?
It is simply the interior of a circle of radius around , where is some complex number, which you should think of as a point in the plane. The radius is a real, positive number. The punctured disk is the same thing, without the center point .
Why is punctured disk neither open nor closed?
But why is it not a closed set? All the points |z| = 1 or the boundary points are contained in the set. Unless they consider the point |z| = 0 to be a boundary point. Then the conclusion to why it is not closed is that the set does not contain the point|z| = 0.
What is open disk in complex analysis?
Open disc: Let z0 ∈ C and r > 0 then, B(z0, r) = {z ∈ C : |z − z0| < r} is an open disc centered at z0 with radius r. Boundary points: If B(z0, r) contains points of S and points of Sc every r > 0, then z0 is called a boundary point of a set S.
Is punctured disc connected?
Answer: Region (a) is not simply connected – the “puncture” at the center of the disk would prevent any simple closed curve around it from contracting to a point while remaining within the region. This region is simply connected.
Is a ring a circle?
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound.
What is open disk?
a disk (mathematics) which does not include the circle forming its boundary. the OpenDisc software project.
How do I open disk Manager?
To start Disk Management: Click Start -> Run -> type compmgmt. msc -> click OK. Alternatively, right-click on the My Computer icon and select ‘Manage’. In the console tree, click Disk Management.
Is an open disc simply connected?
In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that are connected but not simply connected are called non-simply connected or multiply connected.
Why is an annulus not simply connected?
Definition A domain D is called simply connected is every closed contour Γ in D can be continuously deformed to a point in D. The whole complex plane C and any open disk Br (z0) are simply connected. We’ll see shortly that the annulus A = {z ∈ C : 1 < |z| < 2} is not simply connected.
Is a ring 2D or 3d?
The circle is a two-dimensional (2D) shape. It only has two measurements, such as length and height, and is usually called a ‘flat’ shape.
How do you find the radius of a punctured disk?
A punctured disk can be described as an open annulus with center x 0 and an inner radius of zero. The region is defined by two inequalities 0 < |z – z 0 | < R, where R = the punctured disk’s radius.
How do I create a punctured disk?
Imagine a flat disk the size of the milky way, then take a (very thin) pin and prick through the disk in the exact center; You’ve just created a punctured disk (sometimes called a deleted disk ). Punctured open disk around z 0 (left); Open disk around z 0 (right).
Are punctured disks closed sets or open sets?
Although punctured disks are sometimes described as an open annulus, the disks themselves are neither closed sets nor open sets, The disks aren’t simply connected. The puncture prevents a closed curve around the region from contracting to a point while keeping within the region (MIT).
What is the difference between a punctured disk and annulus?
The disks aren’t simply connected. The puncture prevents a closed curve around the region from contracting to a point while keeping within the region (MIT). A punctured disk can also be written in terms of a Laurent series. The annulus is the region between two concentric rings—two circles that share the same center.