Comple Nambs
"4 complex numbu,
Zt2
Z- z
no, -i ckey
Z (coso
tane p)
-j9
poler
Conper fimn
w fc):U t iVcoy)
fomalas
.o.sín.iz fenhz
coshz
.fanie tanhz
e sioh o
" (osho
temho
"aos ine hz 2
sin'z + cos'z I
" Cosh' - sioh':|
": sechzttanhz )
-cosech'z
the folaniy fnchira
two Yeal fumfiras.
) fce)=z
u= X
fce: z
fce):
utiv: z+ aiyt Cy
2
fca) e
e.iy
e
(cosg + isihy)
stiv t esna
4)|fcz: sinz
UtiV: sincr+ iy)
(CinCAtB): siAcosB + tasAsinR)
Utiv siha cosdY + cosx Sin y
u+iv sihA coshy ticaX
sihncahy
cosxiohy
COS
fce) = cos (ame)
u+iv: cosCxt iy)
Utiv: cos Xcosy
u+iv cosX coshy -isnx shy
cost asly
ve - sioxslahy
- sinbz
utiv: sin iz
9utiv:(sioiCx*y>)
) utiv
cotr) sin
L(isinhr ory -(esba siy)
sisba cosy - casbasia4 x
+ icoshasiny
corbasig
7afce): coshz
utiv coaiz)
(cos Catb)- csacosb+ ohasiob)
utiv cosiz)
utiY
roscicK tiy)
utiv cosCix -y
cosbx cosy + isio hx si'ag.
cosbxcorg
: singsiax
fc lge
fee): log 2
U+iv (gxtiy)
elcosv+ieov): xty
e"cosv
X: e
2 2 24
O+ e cosVt e
’ e2uCcos'r+si) e2u
loy fom both sta
cosy
) y : tan
3)
FC): z+3z fod pro va laer at
at
Note
2 : lt 3
fcz) z+ 3z
y3
2
= x?+ 3x-y+ ic2xyt3y)
CI,3)
fc) -e
-5-i23 fcz)
-S 3-8= 4) C½
iC2xy 6x-2y
+ u+iv
t4i 2-4 at f value te fnd
6z +aiz fcz) =
&
part wel the Find Q) 4
Fdthe roal pont
fcz): Gz-12z +3+2'
findthaiy vale at z: 4-3
fe): Gz?-122 +t3t 2/
2.
3+
2
-l2x -il2y +3+2g'
5x?-12 -5yt+3 120
t2
2
: 5C4)-(2c4)- s(-)+3
dcet,-3)
: -12O + 36 t2
8
fumim F():
|-i
at bi poi
Fc):
Utiy
utiv
Cltx)-i
Utiv C1+)
u cit) V
Ue,-) 49+1 a,-1) 4+1
m
20
Cim1
2’0 fCe) 250
(in
2.2.0
that show 72)
sl<spato.
coumbhay
to. anchony
g
lz-al<e where fce:l in
Linit
aachn ompler (oantinuih
a Limit
<S+m
Jitmt
m, fhenu
mot entt. Thnfore he
t fe Rec' z:X+y
220 220 z' xy'riy
y0 y
yo
m,
’-m
thngre
Tm (z)
The fc s lz
ot cobnu at 2=0.
2- X+*y
290 2'
pa mX,
t y2
)
hene the Limit d
nt ew't. Theafre be funetim
Com t
Pelz)
ros) feo: (24
ceuwboue ot
Rel2)
t fes:
2>0 (z)
pele) x
chaig -m'y
Jttmz
t fsfeo
Hene tiit exist 2 tir conthue.
2f0
contuon
Z0
t fces:
Z90
(0umbno.
Pece
f:
124) sT the fn
2 : D,
Lt Re (2)
22f)
X
-XJitm
Deietve
z= 5t2g
3
fe)3(2- ai)
fc2): 3(2-21)*
put 2- S+2
3(s+2i-)
f'Ce):
) 3 25
deivakie
at 2: i
duivatre i-)
() fin (i
.n("(-)
fc: (a+s)
3z2) (32i+ bz)
fez) 3(i2)+
2
s(3-12)·(42i+l21)
be
tßD.
22
Ezuetis CCRE)
caacy
domain D md
cadifs couchy - Premn qubin 7
Equabin.
-Yx
Neresay Condien
ia a domais D, Whse
Panta! eniveti
proof
FC2) =
at her
(t tletaz) - fe)
A270
boytey)
+ i /
fcz+): U
CK+ox,ytay)
CKr, ytoy
oytay tìVoy
AX-’0
Ucy) i(oxtas ytay)
>f)toxg>ooy ( cKtO,g) V
C9))
f) +
pot AX = O,
cheing becomes.
V
Cxgtoy)
fta: t
(le.
-Du
Safp'eient Contm
it cahtes cR
I6)s‘ the
16
Fc2) z
utive x*-2*
Vy ax
Fcr 2 1t
Fz)
U=X
-)
Vy
Ux Vy
tdow mot ttiu
mot
f(z): e 2
-Cx+iy
utiy- e. (cosg -i siny
-X -X
uttv= eoy -iea
^eosg
-e cos
Ux = Vy
Hene CR E2".
2
1) f ) > e it
utiv e
ut fv e (cosytisny)
eosyt ietry
Me Ux Vy &
-X
dy
ce
etosy t i siny
e(ioy sing)
,x+i'y e2 .
PT te trmtin
deivebe
fce sioz
cosx si'y)
uttr sinx " coshy ticosxsishy
coshy
" cosxt oslg
ayesix.
Uy siox
Ux Vy
fre) st2
LOSX
coscos Ciy)- ox s'y
tos(K+y)
fd doivta
fc: ces z
cos cXiy)
UttV CosX coshy - iiaxeisby
coKcosby
-siox coshy
cox siohg Vy - six corhy
Uy-vx
Hene CR e
& te tee): corr ii
icoxsinbg
-(siXeosciyt cox site'y')
Tufon
(oumbauy
200) Fod
Z= O.
) Derive e nean (ovdihin
Aoswn
(z, zz
Tml2)
20
chooce a peth mx
0
-ni()
a4)
foto):
27
(ot)
Goin'otx? +
Koit
(2),
waas
+
- Vixy
tquatng
Vy -Uy
Uy-V,
dunedve
sinhe: sitiD
utiv -ístnc iz)
utiv -(CiA)Cosg cosbxcosi'x) cy)
4tV:-i (isinbx coy - ciny
utiv: sinbx(osy +i coshesiny
ur: coshx cocy
Ux
Nene
Thera e
coshr cony ti cise any.
) cos (ix -
cos fi (x +iy)
) cosz) ) coshz
260) Pr:
ti fc:cosbz ie
fc cothe
U+iv cos(iz)
utiv cosi(x+{y
utiv cos(i)cos (y2t sicI sio)
utN- coshx cosy+isinbx siny
cos hnsin
uycochesng Vy:siobr ony
Ux Vy
Ug
siobrcorg t i cosbeg.
sin) cory + I cos(ix) siny.
i(-sit)casy + cos(ix> siny
’-i(sinCix -g>
)-i(sin icx tiy2) -i)(sind))
(9a) V=tam V
(xty-du log
bo" from logTaHhg
e2u
stso) 2
2 2
cos*
eCeos+r'sto)
e.
rom erponnti
bo tog Tale
Cxtiy log
detvative. itbnd
erpet
onalgte
27
'S 2
Y -te (9,.)
+ i(ay
fC2): Pe (z4i Tn (22)9
h t
a0) Fod the coustutc a, b, c fou
c
ae
xt ay-i (bx ty)
ay2, + iCxg
) f(2) ti(eey)
)
utiv a+x
tecuni) 9 %+ietx)
e
ueosX
U e
osx
i(2)
Ujv iCxty)
Vy 2
t
fo:
utir
-eioy
Vy-eo
utiv: Re ( x o y
fear
f)-*tay -i(bx*
u+iv= xt a -i Cbxrgy
u: xta V: -bX
Uya
Aceord tng t cp tqn
Ux =|
aa -b)
( Let
u+iv: an- by+i
2
U
an-by
-aby ry
ax =cX
a- b:
Hormorie fuetns
ARaal vehed comtouour
cld a Monnh
Utiy be
the Pact -et2 Imtr) Pe ffc) &Im (fe)
U
Re (
()
Rene
e
e
an'4 b
sioce is ham onie tor
+
Bor?4 bg ty bx
fxx 6ax
b ax o
f
1's
31 Q) ST
Constant
given Pealpat
So lx
Since fce)
Ux Uy b
ts G constt. .fcz is
3) a) An fun cion
conslont
alro y - =o
Uy o
Comstt
32)
Comtont
a.
2
(Jutr cost
to .
+ avv
gud
+
-0)-o
By ce
audu+ avd
u du
. (Dy
D
const
const
Canst
Dt pty dt
2 2
y
fc:
cmbt. poo e
iven ur is
Uvy =-VUx
- Ux
V
th e0.
dip pantaly
-vuy
-Uy .
- Ux -Uy
By
2 2
9-UxUy. Ux
tr
. By Ce tan
f: utiv i
Thue fre constat
ha6re
ke
he
M
Vx
cwes hors
-l. Then ee
utiv
harmonie
fher Ako
hamie fanetos
) Venty t t oe &
he whole conplex ane
Uxx t
hanmm.
V: S2ndy
ne)
CHennne cog
272) paooe that
Uxx + Uyy>
is mt hame
o u
38) prooe hmha
Jind te emupondig nate
Pee u t i
Uyy o
hmant
Vy -24
Vy 2x
)
(2xd v:
V ry2+e
i(*ytizrg)
394)
ciox cosny
cosX
uy siarihy
Uex -sioxcoby
+ cioxcoshy
Ux: Vy coSx coshg
*Vx=-shxsdaby
S-stoxiahy da cnxioy
fcosx casby'y
caNsinhy
Urxt Uyy 6x -6x
2
Bx2
-VY = - 6xy
Uy
VSocyda
+ c (hamni
hanmme
Vy Qx -
Vyy
hasmn
Voc +Vyy
-Uy
Vy u
u- f2x-1da
S-ydy
Ve z9-*+< (hanmmse
2
tral sT v:eia2y met hasmm's
Vxx esinly Vgg 4e sinly
haoc
430) P-T the tmtn utxyx?
harm oie hae
2
y -éxy -s
Uxx
yy
Uxx t Uyy Gx-6X D
Ux Vy ax-3y
-S
dy - éxg
3x-0 Vx: 6x4ts
V f3x4ydy v foryt s dn
3yx-yte SxytSx+
) y
-y'tc
+Sx e Charmomie njust)
Vi 3xy
lee
fod te hanmomk congat fenctin.
-TX
U e
-TX
-7X
-atr
Uso 1ecos ay
1e'oray t ~a'e"oy
X
Ur -7e os
-Y
) -7 cosd e +c
(-corhy):
- Ae tosM
-7.
dn
bat corar-eobiy
harmoni.
a24 a-12
-a unatoshnycly
2
put taluega-2
V 2
(heanena anpqtalo )
y-coshroany
Dyy-Cosbantosy
-(or nheAAyel,
(neimome cngeyt)
474). e
fmctn fes.
versiay te yay
e)+eyay.
Ve= siay Cane
sioy laetee4 eyosy
Vyy:enasiny t ely carg t Siny' t cuuy)
sirzgoay
yasy -erosy -einy tee
2 singne+ ae"iiay ta
sinyae esiay
(4
(fcl 2
4ux'+ 4vx
2 lfol'
- 2luux t VV)
feo] a o 2(2 auyt 2vvy)
2
auuxt ¥Ux)tavtacv)
Culv')
,fe»l'
+
)
avVyy
2
Quustalun'+ avey talw) duuyy
ta(uy+ avyy t d(y
Sinu it
Anaig
hammie aleo.
-Vy
dy
a(u+evw't ()' + uw)')
4ux' 4vy 4f'
2
Hene Pvo0ve d.
Confomel apig
AA Mepping w e floy i call tnfond
cnfoml
psesove
Cwer
,2
meppig
2
polar yir 2
2 21
Re
2 =20
Consides a Le x c.
2
+ V= 2
.: U c´-4 c uo hi'ch in a paraboa
a
l- plane
Consen
2.
Danabola cn w
20,
Consden
V: 0
-Ye
when
whih a tye
fhe plane.
+ the lires xz I,
2, &
ing
2
Let. 2
uti: -yt2ixy
2 2
V
24
2
V
2
Substhte
whih te a panslela
pla
when
u: v4
164a
poncbole plnd
when X>o
V= -ve.
te img
plene
522) fod te
2
to mapring
2
Let
In 2
Re'
2ig
Re
2
Th image
- K/g
is mped nto.
plame.
ocany2<1:
te imag t I</4<3,
Find 2
wnder
cef
2
Re (ae 20
Re
The lmmage
1< |z|s3,
m-ppd to
|<lwle4,
w pl
s4 frd te
te te
img
bamed X =1
utiv
V
when x =I
2
t y to uel-y
subrkha the vabe
nhich i's
4 plene
when y
V: 2
cabskht e ral
2.
An 2
-kty-4xy
[atg)
2
u -
mappd
4
Wee
Diicuss he mapra
Let we e
utiv
:e
utiy e e'
e oy tie'iy
v-einy
ecosy
In polw tom,
Re'i4
e r e
iy
cqmatng.
Reg
Re
when
esigD
2e
2c
)e
2c
(iotyco'ye
uty'
e clnc
tsiny:
Ue
nhich t at tve Uar.
-/< <),
mepp
hW: 2
Re it e
Rei
The image
esRee
S7 )
mdlu de mpp'g Wse2.
We e
Rel. eX,
