Complex Analysis
Topology
Draft for Information Only Content Topology of Complex Number source/reference: Topology of Complex NumberComplex Numbers in Complex PlaneUnlike the onedimensional number line, the range of complex numbers in the complex plane are usually described by circles and disks. For a given complex number z₀=x₀+iy₀, take the complex number z₀ as center and consider the radius r arround, then Open disk or neighborhood of complex numbers with radius r, and centered at z₀: Bᵣ(z₀)={z∈ℂ: z has distance less than r from z₀} Circle of complex numbers with radius r, and centered at z₀: Kᵣ(z₀)={z∈ℂ: z has distance r from z₀} Closed disk of complex numbers with radius r, and centered at z₀: Dᵣ(z₀)={z∈ℂ: z has distance less than or equal r from z₀} Radius r = distance d between two complex points = √((xx₀)²+(yy₀)²)=(xx₀)+ (yy₀)=zz₀ ⇒Bᵣ(z₀)={z∈ℂ: zz₀<r} , Kᵣ(z₀)={z∈ℂ: zz₀=r} , and Dᵣ(z₀)={z∈ℂ: zz₀≤r} Interior and Boundary PointsBy definition, let E⊂ℂ. A point z₀ is an interior point of E if there is some r>0 such that Bᵣ(z₀)⊂E. And by definition, let E⊂ℂ. A point b is a boundary point of E if every disk around b contains a point in E and a point not in E. The boundary of the set E⊂ℂ, ϑE, is therefore the set of all boundary points of E. Open and Closed SetsBy definition, a set U⊂ℂ is open if everyone of its points is an interior point. And by definition, a set A⊂ℂ is closed if it contains all of its boundary points. {z∈ℂ: zz₀<r} and {z∈ℂ: zz₀>r} are open. ℂ and ∅ are open {z∈ℂ: zz₀≤r} and {z∈ℂ: zz₀=r} are closed. ℂ and ∅ are closed {z∈ℂ: zz₀<r}∪{z∈ℂ: zz₀=r and Im(zz₀)>0} is neither open nor closed. Closure and Interior of a SetBy definition, let E be a set in ℂ. the closure of E is the set E together with all of its boundary points: E̅=E∪ϑE. By definition, the interior of E, E̊ is the set of all interior points of E. Bᵣ(z₀)=Bᵣ(z₀)∪Kᵣ(z₀)={z∈ℂ: zz₀<r} Kᵣ(z₀)=Kᵣ(z₀) Bᵣ(z₀)\{z₀}={z∈ℂ: zz₀≤r} With E={z∈ℂ: zz₀≤r}, E̊=∅ With E=Kᵣ(z₀), E̊=∅ ConnectednessIntuitively: A set is connected if it is "in one piece". By definition, two sets X, Y in ℂ are separated if there are disjoint open set U, V so that X⊂U and Y⊂V. A set W in ℂ is connected if it is impossible to find two separated nonempty sets whose union equals W, X=[0,1) and Y=(1,2] are separated. For example, chosse U=B₁(0), V=B₁(2). Thus X∪Y=[x,2]\{1} is not connected. It is hard to check whether a set is connected. For open sets, there is a much easier criterion to check whether or not a set is connected: By Theorem. Let G be an open set in ℂ. Then G is connected if and only if any two points in G can be joined in G by successive line segments Bounded SetsBy definition, a set A in ℂ is bounded if there exists a number R>0 such that A⊂BR(0). If no such R exists then A is called unbounded. The Point at InfinityIn ℝ, ther are two directions that give rise to ±∞. But in ℂ, there is only one ∞ which can be attained in many all directions.
©sideway ID: 190300016 Last Updated: 2019/3/16 Revision: 
Home (5) Computer Hardware (149) Software Application (187) Digitization (24) Numeric (19) Programming Web (554) CSS (SC) HTML Knowledge Base Common Color (SC) Html 401 Special (SC) OS (368) MS Windows Windows10 (SC) DeskTop (6) Knowledge Mathematics Formulas (8) Number Theory (206) Algebra (17) Trigonometry (18) Geometry (18) Calculus (67) Complex Analysis (13) Engineering Tables (8) Mechanical Mechanics (1) Rigid Bodies Statics (92) Dynamics (37) Fluid (5) Fluid Kinematics (5) Control Process Control (1) Acoustics (19) FiniteElement (2) Biology (1) Geography (1) 
Latest Updated Links

Copyright © 20002019 Sideway . All rights reserved Disclaimers last modified on 10 Feb 2019