MAT 3730Complex 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

0

0

0

of odneighborhodisk/ open an called is

0 and Given

z

rz-zz D

rRC, rz

r

0z

Example 1

diskunit open an called is

1 zzD

Definition 2 Interior Points

SDzDSCSz

s.t. of nhood if ofpoint interior an called is

0

0

0z

S

Example 2

Si

zzS

ofpoint interior an is 2

1)Re(

SSCS

ofpoint interior an is ofpoint every ifset open an is

Definition 3 Open Sets

Example 3

setopen an is

,0, and ,For

21

2121

rzrzArrRrr

Example 4

setopen an is

23 zzS

Example 5

setopen an NOT is

23 zzS

An open set is connected if every pair of points in can be joined by a polygonal path that lies entirely in

S CS

S

Definition 4 Connected Open Sets

S

Example 6

connected is

231 zzS

Example 7 Re( ) 1 or Re( ) 1

is NOT connected

S z z z

An open connected set is called a domain

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.

Theorem2Let : , Doamin

If ( , ) ( , ) 0 ( , ) ,

then constant in

u D R R Du ux y x y x y Dx y

u D

Idea

0

0

is a boundary point of ifevery nhood of conatins at least one point of and one point not in

z Sz

S S

Definition 6 Boundary Points

0z

S

set?open an Is .2? tobelong Does 1. 0

SSz

Observations

0z

S

SS

ofboundary thecalled is of pointsboundary of sets The

Definition 7 Boundary

Boundary of S S S

Example 8

21

2

1

and both ofpoint boundary a is 5

23

23

SSz

zzS

zzS

Example 8

1 2 3 2 is the boundary of both and B z z S S

is a closed set if ontains all of its boundary points

S CS c

Definition 8 Closed Sets

S

Example 9

closed is

23

1

1

S

zzS

Example 10

closednor open neither is

231 zzA

Example 10 231 zzA

S

51 3

Not open:

Not closed:

pointsboundary its of allor none, some, ether withdomain tog a isregion A

Definition 9 Region

pointsboundary its of allor none, some, ether withdomain tog a isregion A

Definition 9 Region

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

pointsboundary its of allor none, some, ether withdomain tog a isregion A

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.

is bounded if , 0 s.t. S C

r R r z r z S

Definition 10 Bounded Sets

Sr

QuestionCan you name a unbounded set?

Definitions Dependencynhood

Interior Points

Open Set

Connected Set

Domain

Boundary Points

Bounded Set

Closed Set

Region

Next Class Read Section 2.1 Review Onto Functions