MAT 3730 Complex Variables Section 1.6 Planar Sets

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Presentation transcript:

MAT 3730 Complex Variables Section 1.6 Planar Sets http://myhome.spu.edu/lauw

Preview For real variables, theorems are typically stated for functions defined on intervals (open, closed) We will introduce the corresponding concepts in the complex plane Mostly the same as defined in R2 (MAT 3238?)

Definition 1 Open Disk/ (Circular) Neighborhood

Example 1

Definition 2 Interior Points

Example 2

Definition 3 Open Sets

Example 3

Example 4

Example 5

Definition 4 Connected Open Sets

Example 6

Example 7

Definition 5 Domain

Domain Many results in real and complex analysis are true only in domains. Below is an example in calculus (real analysis). We will take a look at why the connectedness is important.

Theorem Idea

Definition 6 Boundary Points

Observations

Definition 7 Boundary

Example 8

Example 8

Definition 8 Closed Sets

Example 9

Example 10

Example 10

Definition 9 Region

Definition 9 Region T or F: If D is a domain, then it is a region.

Definition 9 Region T or F: If D is a domain, then it is a region. T or F: If D is a region, then it is a domain.

Definition 10 Bounded Sets

Question Can you name a unbounded set?

Definitions Dependency nhood Interior Points Open Set Connected Set Domain Boundary Points Bounded Set Closed Set Region

Next Class Read Section 2.1 Review Onto Functions