MAT 3730 Complex Variables Section 1.6 Planar Sets http://myhome.spu.edu/lauw
Mar 15, 2016
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