y
-
D
ENG 30
022
C.
Ieqwu M
6n9inetfin) malH5)
Deforh n of Com pleX 9umr
Forms of CompeX u m l b
repe SEntah on
GCome t
CompUx
O+ Com poX
Confujaks
D fn ith
numlouS.
unbuS
CaM pllx
Las 0f
f
m lAss
of r n s
SRetthm
of
Defhabon
A Cumple x num ber iS anum loe
he Aom of
i
settd
paa f
here
x+
com plix
umer.
CompllXX
ntm las
dhe
part of
hat can
nc
x
sh
+
real
forms
magna
Part
of
Com pux
Part
num bes
Co
carttsian foron (xtj)
(D COmpent for m
Polar form
: r
Fuler form
+isinO)
rerzle
f
mod uuS
ent o-f
Am
E|
quM bers.|2/
olf Xxnumbersl
Conn
COMpUx
ComPlex nnmpux
p
Is heeal
in an
Cieome tm
rep-eserta-h n
f
CompUK tlmbes.
Compllx plant
Z = 3t 4
4
S
rzl=, J32412
A Z- c ta (
NB Prinipal arqUomeat
J25 5
4(Eea) ei)
0
Aae
5
360
Lads
d
of Comple X m b S
cimmurta he laoreespe eive
ae
ess ae
Sul-t
supe
Addiden
ZtZ2
meltph ah
ThS
Zt22
true hur add
add
+Z
Z
22
3t5t
5t8
Z2 t2
5+8
Z t 22
= 6+ 10'+9t -15
Z2 21
+10+9
- 5
ASSocjcthve lq: Irrespeci
ae guptd ha Sant «Su Hs
E
lqo hulds
EE2
243
This
qumlo-
mu tplicatn
hn
Z
houo CompU
dr a r mged, h S d m e g e v m t tiCa
o otono d
S
f
lcuo holds tue for
Adihdn
-9 +17
=-1t19
f
hoo Comp ux numers
wilt
aclelitan
Z Z2tt2a)
Shl
obtgine.
nd mulhe i Cahian
ta)+z
2 t(7tz3)=
5
22X Cx2
mu lthplicahuns
3 Distr) lourhe
22 t 2
7h1s STTHS hat aComplx OV9ltr
numbeSS acld
Can distmibuted Ao c goup of Comp Lex* numbrs
adde
shu eolota ne d
oths ond he sam re sutt o
a
EXamplo
Conge
Z2 t Z,) =Z Z2
Z
nuimoer
of cmplux
MeConjugat 0f
Z,zs
(E)
1S a
compUX numlba,
a
C1S
port
ha
will loe
(onju/gH
he
y
t
C omplux o
re yers
d
ettd
Pre
pre trd
i
Yropes ties
of Cunjugetts
f
CmpuX
numoes
2+Z2
Z
Z X2t i
t I J tXatij
+X) + CCJ+3
=Z
t
=
Xt2 - i(J+
)
2tz= 2Re czE)
t t I-=
6)2Z
2
2ReC2)
2m 2
21= 2lm
z
2
=
(R+m2
D
ez
2
C
tJ
3t2
3
2t
Zx 2
(3t2) (3-2 C)=-G +G+
3 t2= 1t+ 13/
=
4
13
=
De-finiti on
CD Ne ism lookl
ef
em
ho
ofCompLix
mlo
De Le4td njqnbourboocl af Complux
e)open
Set
clocloR Set
ConeeR4 t
intur PO1nt
Ext
numloer
or PO nt
)bounclarg P0nt
Doman
Renyen
(Nei gn oauh 0od
The
1S
s
Conpux Numlor
n1qhbowht0 d
of
df
a
nuner
Cunlx planu
points f r which lz-2o/< S; whare S7C
i s a Constant
mpux
NB. Chec
o at
NPes of
dimsidn
S
ExCIM pC
Zo
ad hooouvhwd of he poit
z-(240)
x
2+
whore &I
uwhore
<1
+-2+0/5I
-2+ J- t/<
l&-2)t i(9-1)1<1
-2+(J-) <1
NB healoove
is cnct oous t
haq
of a
cici
a)+
+(3o<r2
Cent(a,o)
rachus
Cent (2, D
raclius C)
Dele-te
eighbourhOod
d e deleted neghlaouy hodc
i
whch
Pont
Zo
s
of Z» is a negnbow hodof
Omrttd
lottior
otior
such
Poit
hat Some
Am in ttioPorTt of-a_s4
S
neiwhloouwnouu
Contiiaed 1a S
is
a
poht
0f tnet pat 1s
is CompLLRI
CompURl
-t(-)3
R 2 71|
Bund o
ont.
The leounclc
hat
poiTt
eyer
po
of a
ne qhlour hood of 2o
S 5 and at uast
O t of
Oint
point
IS nersr
aSeA 2,
k
js a pont
S
a
o
sch
Conttins at east One
one poit nOE
ExnurpOin.
Ph
ha
sets
SS
lound cid p0,at Ae an itndr
Sad Ao e an exttsidr pout
Pout
open
penxA:- A
n r
which
one
Sct1s
Cotayns
onl4
pora-ts
e.ReeyRec2)71
E/<l1
h n r ond
closed set Contans al qh ntn nd bas
Closed s e t a
ar poHs
R
C2)i
open
e
s
es
ha
exheH
eteH
is
egi on
loourel
pa
Conaeete d
Can le
seB S
A doman is m
A
of
a
Conne ete
pointsI ad Z1
bjapoynomial
set, han he
sets said
Lorlh
SOma
A
pon
S r t c h inq
f
a
Z-2
EXampll qivo
rtei dn
<Y and
(
Sol
Genual p^vaedure.
z
lee Caanu
Cop A ConneHtd se\)
clomai
zoKa
adius
b
-)c
a
Ara C2-Kc
(
an
line mat
set
p e conneGed
n
-2i)<
z
-2<1
+)c| <
-o)t i(J-»)
<
Co,2)CAm
adius
A
2-2e) »
Ccloctsix)
m ticloc ICSe)
NB: hat
7 includes Ahe bounda poit Gnd skartcsa
Should
dpot setcd wrh a AniCE lna whi
sHttuing
COmpux deivath
Ancalhc
fnchon
Cauch 1 Riemam Euahon
Hasomonic nchon