1. .,ti'ri'1 de{irle ( = v-l :}
r,^n
i ij Lo-.p!fa nNmte*- Z rn%
r",-D l- n caj -erJ-c.n3u(ef ]oCrr^ l-
z-- re(Z) : f CosO =
'-r '7) _ f 5,.,-o =
t,
= -Lrr1 t
^-i
} A
,''- ) tL_-L)L
& w riffen
7- v r..r"VL
-
-L -1- t
J
-L ( z +7,)2
+( z- 2)
o[,*t--
fSu e
f e'0
D
.r's'L
LX I" U fiitl'*r, s
3t e t -1 ,
= -L A.nd, L -: l-
D
2 'J orms :-
vFrrc
l,{1
t
_l
,, ,fl
Trr1qrra(y Ct(s
|J
Srnce Z= X+:J
G*pL^ -
r (osA + L"
7=./+rJ
rr f, t)
ftc( &XLs
: r(Gso aisuo)
-) prO"r fo,^
.:
,l
I
a
f
Fi.
':i .lel
Z=fC
h/Pri ,
tj1
:i!
,t
,.i-l
l:.:i
.i;l
.i.:1
t:,1
:.:- :t
I,l
hlr.l
6',i.)
i.: i
:,!1 r:l
i'--:.1
':...1i:
rli.:.:i
i
.i
.:.i
.' , i::. ':,'. ij
l:: ;;1;' :i
-t
2)
-F.Qoo
fo.*.,
;d
= r(Gso+t'sr^c) = r-tiso
2. 7o
2 r'n
"?g wtrA
t
-t.
D
mus-t :af t s/y
-Ct, {eneabr lA^t :- 4P Z= x*rJ = r€
,. fa tf
tj Lcntprex C.t*S,t. o? 7 = Z :
(8
-ip
-)/ ,u.r,r'
/L _LY
J
zZ = !zl
roII
2
r*z ^l
t D
o t'A-, o{t-'aL'n .
J
I )r " 'ne{ail", . l-7-7" ! = d*fant, f'oo ru Qot'f z< ) -/i a{( c#
U --J )
t/n Qot,t 7o b'L f/* Co^flrx PC*''
I
LI
f-n
)r) (7' 7, )
Iz,z, I
=42,
= lzl . tlt
lZ, I
<
17, )
5) (4t lzt
I z,.l
=tZrl
-
'7
-<l
=_-_
-4.2
3. A*
7*_
.. 5rT. L -=-.
zo 4
Jt-
-t (os
-:=>
Z= 51.'
'srn
T)T+L
3 f 3
-{ t V2
=I
J-t
z
^.-'
:) 7-- Cts T
-1 Z= (os)r v r" SAI -
_t
- 't'sl -g) 1= -r I
+ L >t'r1
3 Cfs txlu) + tt7=
5 Ct'slnltz)
s ei
!'t''
...._'- ,/ -
!
"t
tln
a r' Sa'r r 2t'
s) 7= I l-*i )
-) Z=
-t
=8/zt
4r
b
Tn./
6
{,1 3
rl
+ /t 2JP
=
--:L
i I n/as
+ 2{"
3{s
to
slJ_3 t tn
fL
/o
tl
' Or'
31bs4
s' 6
7Tr
6
3
/o
a2<-J
lo
:
L 7z/1
4. a} Z-- t+tE
ifutt +'2Kzr)
z= 2e
t-
2) z= $3 -f
,-ntS* 2Krt)
z=.2e
z t ./_J-/ &
-_ -SZ rr {z
t I tf- rlt, )
a
;(*/,, +2rcrt)
2eZ=2
i) 7= -2-
v -(
Z= 2S e
ds
'lzr- hn-'s/a+ 2rcrt')
-) z= set(o+zrcT)-- 5e'2Kzt
i -Tlz n 2Kz')
>./
j;
z -tt<-
e
i tTlZ + 2y-4 )
e
5
...E
-;) Z=
:)
t'( -
L'llry2Kzl)
e
e 2vr)-tt
7
t
t
T
t-
LAO
z
s _},,f
z-*
,l
Id!
K-- o); ilZft tl, * 2,
5. J JP A)-) rr 4= X+tj =
11
= f C{s(rt0) =
t'O.n
re = rc6u
(re,'*)^ = f^ e'n'
./
,"n ( (asril +{ s,,- nO)
+1^"
j
1,o
2) T}'" r Coots oP egtx.Sto^ Zn= 7o
frurA * fo 0!r,,t
r eto
r ( ,A.g (2"') 1- 2 K t)
11 e po*,w t*fuer,
GE
Q"f +U Foc;hs be 7 =
^ t nO
r'- e
-11
-r1f = rq
Vn
-r:f=
rt0- -agtz.) +2Kz
-> 0= Ag(z,t + 2rtr
I
/^frl6
1,t
r
; rc=
/g(A) + 2K
y7
o,lr 2, )
fef aftd 11 foots are
ffi-t).
6. nlI I :.
-3jD V+[t6rr i -
t, 't L0
r-€ /6 e
i( r+2tczl
:) r)=. 16 ==)
Y-=2 and h0=
:> 0=
:'( {r ry-)
7- )ea--
_ t'!1 tiJ,
rl+2Kn
]T +2Krr ;lC=o,1,2,3
tk" i, rook art ; t( =o , 1,2,3.
r-a -i ?O
.L
-z:)Y=
20 __
t1-L Ctrobs o.Ce
Q"U +" r. q ui.eJ
I - -
-1_
-st -_
2f..xrfste Z= retO :=>
r.( _r/s qQKn)
OO
-) r-- ;
L+2<zr :> O
5
.t _rt-rKn)
s et' -6
/5
; f( = o,1.
7. ]LLL
lr
1-
1+t'.lE =
-,"
1
a. L,
-)-
]otr r,. - (' zT/z o
2e 6c
Vrr
1-.-t<
:!>
6!-:
i+ j 5
-f
2o/
2(ros
e
6 r="
- {'v5 )
2'/
To zr
:---_it
n/9
4
r t
-TncI lane co*,.cr-Ln "I firl = - I.l-t
V2Z -V3 -t
---e
5
nr-}cK.u.T on '. -
.
Qlo.ninoSo v- y&,Lsb nob eq..*Q t" Zeln
B.''t '{T z {s - t
-ts> rof {'5o
= 'fi-r
5a
-F f '= ti7 &
{
V ,/to
f=2 0=
11
---r- Z'{ kl*.!
)
{
so
=a ;B- z'= E+L1/7'E
r.( 7+2Kr1eb
n
t ,./ 1-,L l)rp
:/o { ( SO , -s ,*) ?
2e (
)
i
7*2Kz=
]JV
4
50
f(
l l, [
a nr"xrc! ^^C}.( eqsuer bo t& tl€ pova.rI
,_ _i n/L
= z e
-
-,t tt/3
'2e
,D . 'FV-/L
"Vz/L '12
!2 e /
{" Scrr
r9
2 {s
t
= v2 €
bo 2o
2
&-zs =3
-T. ,
-aJO r^o.n = G^pU, pil^n
8. r.l $t=r
JJ
/ r1ell
'*?rr, ,
= U+iV =
I X,3 )
v tx,3 )
{tnl
a a Co*pD.x {*.,&to",, dn t^,.e Co*p}ux U*rt..b!. z,
= f t x +i1 ) -P e'+
:
=l
ftz)
V=
GN-:
+=
Q. L $s.r)
T * {t2,,)
Ll'+ v'
ag ( {tzl
-.Ui
CTao<- rtacq# ; O Consi4a' St=l - 7'
{^(
:> itzl-- itx+r5)
. ,-) -- ,-t-5'
tte> -- St reta:
-|' H- f'
2
=x+iy) :f-3'+ttx1
Crnc| V = 2,"lri
and + - 20
i
O Const der {t, ), =
o
itxu)v) -_ JJ x+tJ
X -t:y
X'*J'
:)U -J
I
,.1
a
{
ll*-t.<
x'o_y'
&v= x'*J'
9. --:> Iu rois l--'Lr j I L- )
t-_--=a.... . ----
;gt'0
/: ?
+R=+k +=-s"
ts /A., v aZae,
"PT"fiar* eP fi: rzry!-u {y f@ -,
rnder *P,.1 {tz; (s d.R'.,ud.Z__ x+ iJ
-fhe do,n*"r .P tt") ts f"pr.se.h.J. {n o. C.*p}uo pQ*n.
fonsisii^g "P X cris *,& S cr,Xrs.
rc
- .fn:t $:-S:::8g 5"" frrr'
-iqQ,,.tes .? $t= ) , if is Cepr-s5sntecl. r". "tCo*plnx
pQo1,"
- -' L" n
-cnSiStill o{ a1i5 c,'C V aXts.
ts " Csrresp o" Ca1
So*otn
TanrP.rmatt^ frr* a do,,ran tb a fa(1e rtder f/" f'
ra G/!*d *oppg o{ ti^osfor m^h.*, .
.D,-*:r
"l
/he
{r=t
r
--+ 1(z-) ------>
10. Sk fcI^. t!"i Corn,a.iAs
-1 4r,rZ) )o
zi Qt=l + 2 A*rLz ) =. 5
:7 X+2^J---5 " st, Q(n" "
.sN
I
2). _ 3J >6
11. I 7- 7*l-= a to
: - -.'
a{t eq u*t, ) " P e ,'ra{i -d-H* (infr.
^F/
7o qrtd {aCtus e.
s] ZI 1z
5l
I zi-tl}J
r jTl 0g(z) 5
2 o -,< agz< t/z
c
,
s) i? <3
12. _i z o,r,'-1 l| ,. -.2 +nl-l - J
a cirdb of {^or)tus { fr fnfr
af z= 2-L;
L-)
l:, Z -l- L"(Z
"gr4gA 14t,o.nd
^J
-: J t,
-l
-{
;-)-.) l.ll-:_ -l <Agvt
2) { (z*;1(:
A1a/r /l
- -:-
LI
.) 3n -7-q
13. l,
-i,-. I /j!4) =
Z f€rO
'Etdra
f-/
r
3s.
O
L
o
_{0 t'4-
Ke
4l=
,
c"nrain
:-- r
r-tueorptr :-
ii l
Inde.
solo'*:-
x +t3
=p Z=
/-U _t.
2
l-v
2
V
2.t2
Ia
+ , ll
unC. Te ,^OS* .,f le rESi 01
r) o(R. tr'z) < l
C
f(z) = 7 27
We .o..r. V7 /-22
(tt+t'Y)t-
14. 3,?G;*,_
-s
L/
-'/ w=
',Y= f "
llnder f,rn l-.ont {o n* o*,d, /") fr., =
/'/' ) {, a,, =
tfl
t") Y{( haw N = Jt zl = z'
(fg = r- g -='tYe
tr
&
t
Trr-e {-e q t'cne
I
V
- -/ I
-t
,/
o<, t d i s J
2
Z
I
7
P I .ro
s!=>j<
r
-l/
r?317) = Z
2rZ
3
(
/ I I ---.-
7 l-- ---
7
i-'
15. L'; wrho-ve Q. (rz) =Q. (
h
-> ,,iYe^ o( -I <1 +> o
-->
{,=,
->
i. 2R(z) 3Tn(7)
,2
L'K -y)
,2
.'
rDt*ff
=> 2-2u r 3Y < t2_r
![.,ti'ri'1 de{irle ( = v-l :}
r,^n
i ij Lo-.p!fa nNmte*- Z rn%
r",-D l- n caj -erJ-c.n3u(ef ]oCrr^ l-
z-- re(Z) : f CosO =
'-r '7) _ f 5,.,-o =
t,
= -Lrr1 t
^-i
} A
,''- ) tL_-L)L
& w riffen
7- v r..r"VL
-
-L -1- t
J
-L ( z +7,)2
+( z- 2)
o[,*t--
fSu e
f e'0
D
.r's'L
LX I" U fiitl'*r, s
3t e t -1 ,
= -L A.nd, L -: l-
D
2 'J orms :-
vFrrc
l,{1
t
_l
,, ,fl
Trr1qrra(y Ct(s
|J
Srnce Z= X+:J
G*pL^ -
r (osA + L"
7=./+rJ
rr f, t)
ftc( &XLs
: r(Gso aisuo)
-) prO"r fo,^
.:
,l
I
a
f
Fi.
':i .lel
Z=fC
h/Pri ,
tj1
:i!
,t
,.i-l
l:.:i
.i;l
.i.:1
t:,1
:.:- :t
I,l
hlr.l
6',i.)
i.: i
:,!1 r:l
i'--:.1
':...1i:
rli.:.:i
i
.i
.:.i
.' , i::. ':,'. ij
l:: ;;1;' :i
-t
2)
-F.Qoo
fo.*.,
;d
= r(Gso+t'sr^c) = r-tiso](https://image.slidesharecdn.com/1complexvariablesandfunctionselesra2ahmedawad-151030173028-lva1-app6892/85/1-complex-variables_and_functions_el_esra2_ahmedawad-1-320.jpg)
