ace academy notes of complex variables for gate and ese with examples

complex numbers

1. COMPLEX VARIABLES

2. Complex number :
no a ow *

ÆNcrf complet mumker can be
represented im a form 2: KH -Cay

whee y are real numbers.
L£NEr] com
ach the

eL mumier 3. X+J make some emsle xe)

post HVE dueclion 4 X- axcy called a (08)
amp le bade La Complex umber fre lr Vay" à

called modalus (ar) absolute value A a complex Be,

3. Ficoding argument day
From the cag ram sinb= 2» 66> A =>70m0= I
o Y x

36: Tom! (4) y the als Fae:

pler form ue hove 4: x4ly
= HCAOET SEND = ¥C CBO FSO?
- x el

al ere? 5 Ic paler form F 3: 24

4. pta nd ‘i dy ba
comfez mumier ¡hem Ihe -
y $ 4 denoteÀ by 3-2 re)
proper tes € miles (4 13-2] 7 141-181
4 AE 12, (à) # lar att lanai’: 2( 1st lJ
+ |4)- a it 1240 A ay ica
33 = GAY

A + a; < ide asl AN
lar

5. observahions on aime a)
a te

ars CAY) = Tan ne

2

7
Here Jane = 32 Jane= !
= 79 (9%) +1
algo Ton (27+) = |
Tan nC amt Th) =|

Tan (enmF)=1 dos W20.2,-°

6. ade Oh pou d a a a 207
you Le Ep 4 a e A
a fal 4 3 5 ag ds BtanT for m201%
Cc A SN
Here 82 Asa is called principal value 4 any 3

Generally [1 < ang 447 |

7. oherva Kon
PATAS

Ag CH) > Ars 01)
Tond= 42) Ten 6-1
>) 9-1
: 4
a)
«y

. @= Toms! (à)

Ars Citi) = AX Cu)

Ton 0: >"!

> 70m CA) = "|

49-1 x nt
correr

" Fad) (1-4) 24

oe Ton! (1) +7

EN ‘

=, +

8. * Ars (x4%0) 20 tf x70
iT if x60

# ag (etiy) : FY yro
Mp Y y<0

# as (ot!) = mt port ble

9. &) principal value 7 aquel € yee

22 -V3 += (ERED =|)
A Hore

= TO 2 =
E Genera] value 4 agament € 14830
sh Here 3- I+Bi= C3)

General value E ang gs ras! (2) +297

Mat any ! ned215-

10. >

*

proper fies € ¿ encia] value q argument
A CB... An) = E ~~ +05 Ry
ang (2): ag 4,- any à
arg ca) = naná
ay Ge - 21 Ca)

11. # AT Cay 4 dm) = ABTA A - - HG bn + 207

for n-où-l Al

Lis” ee Ag AGAR for me OAI
4 Ag a= KAng CA) +2nT fos n=0 a IA
* AS a - Arg (A) +27 for mo &I à -|

12. @ The Value 4 (y

pr 4- Hi set. zu a a
ST =
Wen Gti) E (e u“ e- rañl(+)= We
oe] oF.
ars
= 254 [esar+ isto 4m)
= 206 (1409)

AS (role 258

13. ey

¿y : dy

443 = 2 2-3 = +9

X= Ara née a
EN

| Real (4) = ara ue)
a op (2) = 33,
(oR

14. observan
OA >

Mala aA. chat sim : oe esa -(són 3
—€) 2

Ot@ 4 283 ky ta Lo A ¿A aising

a ta, età = AS

jee Gases - lgmas € m

| 2 | | m
LI

15. obgervahon eo.
AALS ia kta stamilasly
e

# simi) © = AL + ca sha
2 ne
(À À + Tan 3) tom ha
= "le -&
* 4
O AK
CS,
„.[ser@a) 2 Soha

16. Ho +
= c&ot (seno

observation
t+ompt e
Ke ZI for ne 0y1,3- - = Iti@=t
+ (anti rt 2 my CSM ET
A GntDri for 9 20)0123- - ee
il,
FE == + is_nn
Hal = -1+ 1 @)=-1
+ = = a Per Et
e += com) isto Th
= otic: à

17. &

e

* ®

iit

RE hen the value & it
ip

>

qe 7 (A,

Ma
ae
ae

Fem he real past 4 Pa

qt
4 =

ROLE „sr
“Ug 4)

ot = ase es fe Clgo - is do)

Now veal part q 44T = ase co list)

(aa = m (34
O. cas i900

18. . » q es )
Ægnaken q cirde im complex form safe
( 2 AR):
, E

he egueñon 4 a circle wh centre ee

4, Au) with yadius +' cs

(2-2): Y Equation $ cirde &

S y
volticafion : 13-Z=r @-ay't -W > Y
> [@+1y)- iti) Y
> lx) cy-wl=r
4 SENSE
laa VA (Yi) 2 7

19. Ed 13-5-5)= 2 13-5+2c)= 1
= |a-(3-2D]=1

En [a+] 2

2 |2-cstsUlza all

20. ct al ps

€) = WH
a ga)

À an)

21. Tf 2- Y ¿CAD im 2- plot they fhe complex June
corres por to 2- Atly can le corten En a forn
fale ario im W-plane. Herr UV art Yeal valued
fomchon, $ Loy
& $: 3Y- Hay”
za)
a v

22. ED Ya: lay (ED fay: sing

23. ED Ya: lay (ED fay: sing

ma ie
: les ore” ) ie = semCatiy)
= log 74 (ge = Sina cH CE) + BE sim (13)
- lgr+ie = sina c&hy +0 x Smhy
> eV AA oes
E lg Cat) + ron (4) u 60]
0 an

u

(374 = cah2
sim ia: ¿mba

24. Periva tive d a Compler function

A complex function fa) ès sud to le
. . 42

difer embiable at 3za |
Le WIKI exist. 74 is embed by Hla)

da ¿ua l
(a: q Be ee Usp nou
A Ja fur) „ U- AAA (a: a”
ler Er 7-0) on ET ya) > 22
Le (24010) Par - aux)
= (ri) +0) =24+2
= 2+2t

25. . D
& fa: x5
Brno ale
£ (a): an”
Ihre {'(2) dees mer exist at A=2-
FQ mt differential at azo bot US diftrenKable eu
all values opor hom 4-2

26. Note :

* fu): sina
fu): BA are diff ren hioble evezy hese

£CA)= eA
a

fa) = €

* polynomial fonctions are differentiable e veryuhere
£a: fu): 143844: Ya): 242 de

27. cauchy- Riemann egpations €<-2 egpattons)
A complex Junc Hon Y) = VHD & differen Hobk then
a and © sabsfes cK egpattons
ce Qu = Sy (u,= Py)
and 2% - 90 Uy = - 0.
a ee
om amd d Va By, Dar By am

No: If 0,9 gakgfics c-R eguah
[28 differen Hoble

comtinuoss fhem ft) = WTO

28. Ga tm: asa

(Ly) 130)

(039) +! (223439)

u v
U: 243 : Vx = 23

Dy = 3 5 Vyps 2x43

Here Un= vy

amd Vy: - Vx and Ur, Uy, tx, by 7%
con Bru 004

at every point m hole complex plume
> jar: a & Aipprentiobe everyakert-

29. Fxg) Ya): àt3
ED ES) ga: 2(3maz)

= a-ty+3 c
ip =
v a oy
Vas 1? Vaio Uq=¥ : V0
Uy=0 2 Bye A : By 22

Fer Unt By at all points on were c-2 egpalions
salts fied ar (09 on

cohde complex far
. a E . 43) is diferenk able of
.. faj=4+3 5 moler differentiable a soit €

30. nok: 34 fear mio yy diffembioble they
4102) = U + dx
2 Ugtt Uy) [- Uy =-%]
2 Bytit, [o my)
CR eguaticos in poler Ferm
4 49. ano ino) & Aiferenkoble then
wee eo (Hs +58)
Up: -y Vr (2%: -r à)

31. Amaly Ke functor
A complex fun chon fa) Y said to be

analy fic at gza it { da) &
differen Hable at every pol within som
d- disc 4 3-9

32. Examplc$
EX: In previos example © LISA hb

{az 3432 à dyferemkelle eve] where

340) à amaly Kc every were
cd do Ya) DH fem

obgeryatm: Tf ck espakons sabs hi
where omd fence ft) &

fa) à differenti ale everf
amaly Xe everyushe7? :

33. E12) m previos example O
da= A+3 ty nowherc Aferenliable

7 43) [63 mo where analy fic

ED In prions example ©
a): achma)
Here $) y Ayferen Hable only at he
origin det met Un 3- disc H origin
5 #9) & mocihere analy he imlodirg or
Cow), bar ck ds AifferenHable at co0)-

34. NO

fa): sa
$A): a
fu): «A ome amaly he eveay where

#0): a
ia) = poyrommial famchen

35. Key potas
# analytic fumetions are algo called regular (en holomorphi<

func Hons
x A fume Kon which y amaly tic every here m cole complex plane

is called entire fanction
+ Amalyfc fanchon is infinitely differentiable
# Tf tig are analpfic gunchow they 143, 17, § alse
fic fumchons
* pmalyhic funchen is alow differenhable bur differen Hable
funckon red mg de analytic:

36. observa Ron
ET a
x
f(a)- e ze ICE & amaly le)
ee
cet Coes ti siny)

lex Urt Uyy O

5 u_ Su _o CS
be FUN TL TO
A Y y

stm lary Var Vyy =? ) ACTE
clearly W% sah fies op (acter,
espahon 8 -o

Wand Y are harmonic Yumchorg

=e chy +i eTsiny
un <<
U D
Y Lay: Uy: = * siny
Aa : Ac&y à: Uyy = A chy

WKY If yrg-o mem Bis called
harmonic fumchon

37. Note. Jf ARS UA iO à amalyfe Fm vuz0 and 0-0
4 vand Y are harmonic funchens amd ‘v' is called
harmonie comjug ate Ku

38. 24 fa)= U+i0 y amalyhe and if u à give
BPP = ae i a)

=> Fimd Var Uy
> 92 fonder fra Au
fort
yan Sen
KH): Uti ds analyke and | VEAP
m m —LÉ nn
> Find Vy vy
=> Me S ryder hr +k
+ y cont gern SA

nets
70 write Ihc
amaly Ke fumch'e
far atid lo
kom € 2
pat x=3 and y=0
cm u Y

39. En) For am analytic fumchioo fa): uo, f Me veal part
Us 2x. find cts harmonic comjuzar Y
sl Gem u: XP TA co
Un- 2142: Uy =-2) 70 wrıf fa) mkrms ÿ 2
Pst X=3 and yzo im UN

ale + dy +15
Now) »-) y Az Sun 7 Ur a 2a: peotork
"yimt 175 .
witha Nou) Ya) Ut

3 ve Secada +fady+r = AMHR +61

40. Ed AGE amalyhe famchen $12) UD cf Uz eX siny fond
a. Algo wrile Yıa) teams f 2

A Giv™ B= e*smy = u: fetcosydx+ fo dy +
a

= er simy a IAS

Vy= ec3y por f(a) pur X=, 720 tn

Wes uf By dat [= Vx dy + wand Y m
PG hon te rer... a. e+e! V0
L sito Now Ya): ui

= 4e

41. complex ines vatton

HC ig a path bekweem foo pains
AtoB If we define a complex
funcio» along he path 2 from APB,

fem woredeme by 108) alorg fhe pac’

At 322+iY
da: dati dj
workdom by fc4) at 4

8
E (vealaa > | gcarcar +idy)
À = fC4)A4

e

42. note! 24 fA) 3 cmaly fic hey (kana $ agerrienk rks

hen { #2) 44 is Emdepedent q poh e. z

à A divecHy amd %
Im this cose imtegrofe fc) 24 $03) & omalyhe
apply the limibs- Ira along all pbs

A is Same -

43. aw)
fícaraa lisa -3 007) (daridy)
€

en)

y
: | rm 87] arian)
Per)

y
= | [ddr da )
o
. > !
el 3), +3( 2)
DA 1 21-8

u

ceo Lime SYA
dy da

44. e of the integral a Gy)
(ara az | MESE)
a

(0,9)

curve y = x? from (0. 0) to 1 +i is

: \Grrelonaanel!
42
=f [eras 2x2] (aut oe]

Oo;
7 [aria de
ö - ains Ax +2 aa

424 #)

45. FRE ETS amalyKe > (saitg is Independent q PRE
[a

ee mi , 5
| scaras: Swan: et. G4". pit . aïe
aro 6 3 3 3 3

46. comp! ex pouxry Series

Lautemt Series BY Ha) & amalyke im am armada,
a

fhem Ihe Zowrent seriy 2 fa) at 3-a &
faa)= Zayca-ay + Ein ca-aj”
mu

(2 by $A) 43
where Ay =: a rer (PART) Ds “mí e Nt)

Ti<13-a

173

47. obscrvafion

Fo

[a-al= Y

48. Taylor aus > 26 year à conalyhc im the region 121er

They [he Zement Sn & fi ak 72: a [23

$03) 2 ay ca-ay C Bn this case by ba ore 7
np

ike en

= AA
where am: = FE a3 I In Ca-a) 4
i @ @-ayıtı , 3 j à
ag iene Cal = ea)

ty a, = ha ae gm.

21

49. a LA
From D 43) = E anta”: a+ a Cara) + ag As

A = sco) + El
2 Cr les Fe) (2-9) + Cm CA ¢ END) (a-a34- - >
§ > 3!

This erparsiom is called royler seriy 4 fin im the
vaio IA-alsy af 224

50. Note ! [69 maly -al7 Y fu mt
a y
cc 12 al Josh
sa Kris

4 Mea abst da
= s

§ (8): Zs 5?
(4-9
«heat eye = ate PE
2qi C ei
M4 ¿-aj 1

51. LD £ ES .
E) Expand qa): e usieg Toy lor Serres ahout- Y) 3=0 dir 471

Bd Gwen 42): en 340) = 0°21 ge: €
dien: er L'to)- e see
gay A gel ze
piteaye ui AE pears

52. (i) Now Toy lor Sex ces 4 403) ab 4-0 &
fen: $0) 4 9 (4.0) + CLASH as

E 21 7.
ne thar Li a

Be ein +4 + a gs
naa

R'o-c és 13-1< ©
€ hesion & ewogent)

53. UN A+ 221
(dm: F0) + eo Ca) 880) Cast OY Case - -
E bu 3!

A 3

AC A EEE en +5

Ro-C & Jaleo

54. —

Alternate method ar 32,

per 4-1: € 9 3: ltt
Now fCä)- ce ES aan = de
€

ln re |

ze 3
[+ Eno + GW, C1.

el 3%

|

55. some Standard Aossemt Series erpanstors ar 2:0

#4 eÀ: más A - poc u 1a0l<e
re
= 2 32-0120
an. - Roc % |
* SINE: 4-93, 05 34. - ROC 8 Idee)
cH Ss! :
CÍA: 1 ah, 24 _ 26 . goce 1421<%

DTA ar
# Tomy: E Lo : LAS 13-0147 i
ah

56. gun: à a 4 +2 - a, Lee Roc à
12

ye 4) 23. a ET poc és 13-041
= (-3)7 = 142,27 2%. Loc ts 14-olej
poc (aol

Roc & 12-015)

pos Y 13-01<

# Fer ([phone] + - =

(majos 172; 44633
*k ca 20-3) ERES ESO
x ara 122432484

oa

57. O
Erpamd fa): A asin Lear en)- Cexte§ ar Ci) 220
G&z 3

4a] NA]

a [4-2

}
MN E 3
442 2 (144) Roc 3 12-ola

4

58. Oj) PE 423

pt 9-3-4 93 4- 143

y /

a — a lo a e
Hi aan ES sE)

at dl e (Y ES IS es
a+ |é-01<
a AE

— 182121
= 12-31<£

Nod Kia):

59. D Expand fa)< Sn at 221
a-T
a PH at=t > ae 447

Med 403)= simá = = Sin HH). -Sint, 1 Es bso
$ aa E ca t(t-£ iste |
2 + ET _ (Bak Fe
al gs!

- ja) Ea: am), |
ee ka

60. ER Expand 18 = “ =) LZaurenf Ceriy (NM

37 3242
ci) 14171 dh 131<2 aim 1<121<2
i J
, 1 | - —
A Given Ka) = Pas. Sy 3-2 4-1

OG) In Roc 17171 x Ay: ==
HUSOS NENE a

J

/
Aa)" u)

61. ah Or ay tO

Lt E+ GP Jt ght ra

Uh In Roc .} ares kaye Er 35

ell Si
(AD “abra
SY 8-2) uy”

BCA) = “3 (148) + (Givers }+ ciara - -

62. Gi) Tm poe Mareo;
a A a

AO
Fo a) = 4-2 Ad

a =
A Faye 30-3) AC
) Kia) = 4 {:-37"- 46-47"

3a 2-4 fr +4 ya TE (1 Gt (ays --]

63. 1
f(z) = E when expandec

series around z = 2, would result in A Aa: —_ 4 A:2

ith the region of

por Bart = ax tt?

The coefficient ax, k E i = 1 = Al
On) FLA). —— + ae
expression = A) fe (- +2) Cal
(a) (1 ji Grou!
AA TE

=) f(a)=- ['- t+ CZs e 3
2 - [a a]
AG) = -1+02-2)- 2-34 apr -

Ar EEN

64. a kK
Gum LA) £ ak Cda)

¡ezo —©

ape cy El
From 050 %

65. 4e "lg
A Ka)-

y 4
Va + Ca DRS ca), --]
+ ar !
atcay= à {md

43 ar at J J
31 ER a
F(4)- 4 41

e

2
; 4
u nf
sa senguler po
zo

Here 47 ;
coef A a = =
i
coef $ AE ae

ch
Corre
ca), Ch), AH) 2%

. ophons

66. SA FA) = yp 2 cm |ati<)
pyr a-1=t 9 Flee 166

te leg cay = [gU+t)
2 t- E, LN
3%

„fa (4) Cam” (AD? ant.
2 3
4

67. 6. In the Laurent series expansion of He x ade
dos region sl Give Yca) 7 Gnas = 37

Im ROC (513142 (/31<2 amd 13171]

z D Be
ET ah)

Zn AG dC"
1:4 (+ CB) Era >)
“Z(G +4) J

Conff d pe a]

68. fesidue : IE gra % am Kolated Simgalos polar $ 42) then
(PEE ee coef q tao" Im Laurent Seras 4 fa)
ab Aza
a = mon Be sour por: If Aza Ey a vemovalle
Sémçu larihy (nen fs fea] 0

¿za
@ ja: 53
: e
Hen Alo 5 am iso (ak A Singular pont

69. Nous Jıal- SYA _ 1
2 Él ES

this expansion confaing mo mega pue pose as & (49
“ aro 8 Yemoalle scm alarily

70. pele 4 Ae am Bolaked simgu lazikr gza & calkd a
ple order oy fe Laurent Sert §
44) abut 424 comtaim negative proses

apto (4-0) *,
£u 403) - al
a) | . pot 4-17
Here 41 & ¿solajed sogaler pon 2: I1k

HE _ ¿er

ns fu): E 7 LE: Lo
A) [24 er ES rt )

71. 2
ha) -
2 La
3 ES
El
e
21 a
ao
=
mo

= fe
HE
Ze
ca
el ae
@
ek ie or
= fe “
“5 | ‘
. 8! ca
a ; D+ =
ql ays
Y .

Nofe! pl & Ed
€
order m
4 4
iS
8 cale
fe Y x A
e
pole.

72. Jaden fication 2 ze omeva ble sing arily and cod hou
Louvrent series
if he given fonction 412): we
3 1 aa y om wolated sims alavity wih multiplicity m'
If 9
cal, Tf $la)to hen da à a pole 4 oer m"
cased: 2% G64) =0, Pica) =9, $'ta)2 0. Papo : compere mm

#24 mom they gra is a pole $ order m”
* A men Men A:a y a rumovalle semgularih

73. a
Ed) Ja. e +23 . GCA) Ed fcar- 243. da

(4-4)! cart pla) (34) (arr) PA)
Pere pta) 4o amd 615) +0 Here dito amd bn 40
4:4 isa pole 4 order (0:0 asl, 2 or poles § otr

Hr

ds ga pole q order 4
be 42 1-2 ase simple poles.

74. & fa: sing - és) SE) yar wa : 62)

3° PCR) 302 PA)
A a Singulosily bias Cees Gi ct Singular pdt

u win mai
dia): sima: — Ko +0

62): $2 9 ECM)=0
gim cha HN: ¡+2 3021

ges -sma > 60m) 1+0
her m>n

i mm
4:08 a ple & oder mon: 3422 L Be Y ie moral Singular)

hese nel

75. Exenfiol Singular point : pn Esolated singular pomt ara es
called am epentiol salat Y IR Aouremt Saris G fa)
abst Ara comfaims infivile nemer € E poux £ Ca-a)

au, ta: Sm(4) G) Ca), Oe ca
his expansion contain imfinte member E neçahw poes
q (4-0) .
320 à esenhal singulorih-

76. Residue’ 2f ¿a is am isolated sirpulow pone 8 fa then

| residue 4 vn], 2 cacfficient Y Ga” in eurent
za
Series Y fa) at A=a

calculate 2 residus
a

Resi due of aemoy able simgalor pont . 2% dea MA certe

sen alar ly hen (sé), >

77. ca): SA , dia)
a pia)

Her 4:0 & singular poor ah m=

dia): HPA => £co=0
: 20
$ca). 257052 $190 pesto...
2 SMA
g"ua): zung + $01.240
ale

MEN. ao ig vemovalle siwalen y

78. pagano of pl ere
ap Ava pole of order E fan

Note!
A

Y“ 664) amd 4
pla)

ee
(res 02) e 4 Abel (ea) 4=a wa simple pole

20 (en) ata 27
# 26 da y a Simple ple len Ez

[Sk Vaca = 4 El (3-1 f{@)

Ltda 2 a C4-a)* ga)

hen [res fan], = 60)
a= a q da)

79. (E) vesidas E fray: UL „at & ples:
¿na
6 por), ‚tr Lea

Here Buıyto : fondo cita

© der ds a pole of order 2 (silo ple) ¿e da aL

1! asa 48 4.) (247
22 wa pole ÿ order 2 (422) me (Ens _ 209)

“2 a”
pesgear] . À Cade TES) ES
Ti u | ) 1-3
ee pastels, : =: -2
al Ena)”

Zee?

80. EN regidue € fia). CHA at 3-0 YAM Has da). cad

a

wir) soa
Einen fc): «+2: DA _ dia)
son ea) y re Cal: 02

pd): $673

Here Azo is a simple pole ar ca
es 4.3) - 4 (2.940) ves (ca) Sa . wo
Í Lo 40 [ lazo * o) cdo
z=4 2x eS - © Eu

330 sy o
HA
LH. wok yegund

81. Residue of eenhd simeulancly = coef E ie
Sorig at 224
N
Be ale e ‘le fe Ca) | Car” ane

+

I! 2! Ti

: J TT Do
O0 a> at 3s

Here 4-0 ty am egenfial singulsriy
[S$]. dl cat: oof 4 L: 1

82. Simple € closed cure F

vas
Y a cled cure imienech isch only © =
firme Ihm fhe curve à calle d Simple
cé cure:

Simple clad curves

© Sy

mot semple claed curves

83. Cauchy - Pesidue Fhorem

24 fa) amalyhe thin a semple
clogd curve EN (Hraversed im posibwe,
divechon ) except af Some isolated

Sugulor points a1, %3- me Ff Yi, Va > Im are heir qescduss

then (HA: ant Cri pat m

<

84. 27. The residue of £(2)=— ze? at Sl Here F=38a pole 9 order 2
a ¿e 4: 3 8 a semple pk

Pega) _ G- Ca-3)* Fa)

3° 333 23
A) À =
IV year (34)
ae el

(6x1 7

85. 8. If the residue of f(z

SA Prom ophin co par 421: @:0
them fa: a _

cap (ard)
3:1 ba pole q order (k=2)
3:2 ly a Sèmpk pole

11 42143 ane
LL y A4“ cos . 4 (amen POY
EN asa ap [ a) am [ (dr? | =

7
Lal) (0), Ce) ae covacct

86. 29. The residue of f(z

ol

=>

= 3 Kr la ¿
here à ? is a 3 ju point
win m=2

BUM) = Tray (Ma) = Eto +0

2-3 8 a simple pole

/ ca,
an: E. Fr ae
tal" m a)

2 TZ
Ast + Geo
= Te

87. es) : U Carta)
4 2%

2G 4x2 > +
a ays ES . ca) 272 ie
. SA (2) 2 4 Rett OI (yat
Cog

Here Blo) #O
+ do Ga sémple pole

Bay sea]
gg) = CBA

88. sA fa) = jeer

. MENA es, Gail oat
pal I- (+ ar rie Ta?
He ]

a4

Dee 4 a2 Ar
ie 7 35 Eb pá 13" o
)
psfan,., TAE
= «eff 4: 4

89. sin(z)
is

32. The residue of f(z) =

A Herr 3:0 & zemoable sing ulavihy

Ban

90. 33. The residue of z.e’” is sa $A) = aca

ETE

2 ALLI L+
Att t ade 2

4 r 93
(3 143,2 => 3
ai bres tao - caf can”
= coc & 4
- 4

2

ata, al MITO

91. uni cirde 1a) =1

a) fa): 4: 4

1-1 am) d-)

dal y a simple pole
besten], nr =

2 Cana)
opta) ts corre eh

92. Es amalype everyuhere( cc) Pr $ 3

b) pak Ana”

> no rescues Ces impide 0 her 4-0 & A Simple pole

oy (dy raiduc hesrem Úg imgide “Cc!
a-oxL
fra: air sam J vida ee | +. ir
€ img Es” =)
ophim (5) ix correct pao BY yesiclug hsorem
(443 =2 0x7 =2mx)

G
et
7 ( 4 OPH) corres

93. ar integral th fe? as faz e"3. 192, way, CaP,
ER 731

e unit circle in the

(GATES thre 4:0 & à singer par
C& dag de c'
mn en = coed a 3 =}

© By yi due peeve
5. fea da: 2x7 = MA |

a < am

= Ja

94. 15. If c is a closed path in the compl

dz is

-3*
Here fea) e 3 es amely fic every here Gy

Ae) cohole complet plane -

See o yes (das emside OT close À path: E

Grasa = amix 0
2 = ©

95. The valle of [2 ES £ € Hor 424-212 are

Simgulor points Ces

curve ‘
outside €
. Sener. dl
sel Gwen circle Y 12-212, I regidues impide €
are ¿ero

By residue heorem

(ca) 430

=

96. The

value

y =fresfıa | me ES)

2
of INR 4-

a,
axe Gray AaB
5-2 (trey

= (Mette = si-4
Here 32-06 &
a simple ple Ces

mside 0

OS Ihres)

(fanaa = aqix yy

¿E = un (6-4)
- 4mll 3-2)
= 4G (2k- 3)

97. 8 he value wh A 2 + q a 31: L— = &
in y Wa es =
=Lis ur
: EE Al
@) Ai $) 0 >, ¿mL [i a E
{c) -2mi a >
=1f14) - ent Fat 45%
eee) = frs) = cold +
420
Lo = pP
2 = In. AL, ya Al) =

+2: E
wy 13% yr Cr" Ey residue Les

fos aix y

y LA
y 2 zarix)= -277

98. Let (-1 -j). (3 - j). (3 + j) and (-1 + j) be
the vertice: rectangle C in the complex
plane. Assuming th is tranversed in

counterclockwise direction. the value of the

counter integral $ is (GATE-21)

faye ——

4*(a-4)

Here 8:08 4 pole Y order Y les
inside 'c'

Her Ket

99. 22. An ir 11

circle *C* is

If C is defined as |z| = 3 then the value of 1

is

(a) -zi sin(1) (b) Zi sin(1

(c) —3zi sin (1) (d) — 4mi sin (1)

(GATE-17)

3 (a: EN
Cri X2-Ù)

Her arta ore Sim ple poles lees

inside °c! 3
G- C3A)x Gye
vie [ves fea) LE 3% Can) (aD
= 2 xl - „ziel
Be L -2
E

100. path C in the complex sá ph La) Es
——

1 in the figure. If

rn am)
Sa OFA is_ (Rounded off to two Yer 4:18 à simple pole les inside e
d e
= Ari
y. E cami] "7 HZ 7%
4 C4/+1) (44)

e ( falda = aix C4): E
= —O0
Given (fa: ATA, 9
(2
Frm DAD AZ

101. The value o

curve c. where c is |z|=4. is

27 (b) 0

along a pot $ 3)- ESA

<a4)car2)
2 ar Simple poles

Hem 82, -
Cig, inside E
_4
_ 4 (ara
= a = 242
4, {res fc la, ER Gr G2)
2 4
3 —— —
vs [res Bt]. 9 = & oY GH) Cys)
2

Sócaras = agicrtrs)

2211 (142) = 20

102. Tae ps $31)

3:4

By residue oye

(ete ar x (+72)

€

=amxçiet-iet)

2-2 ¿A
x (ED) yal
al
Ani Sima)

u

261, i, id
Er lá
2
rel
Ze
. 12 -ia
Sim 3 = ee

103. : ue [> Paria 4 { "ar =] 7
By residue herr om
eee x Tr

= ımx GE) - =