Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
A comprehensive overview of functions of a complex variable and calculus of residues. It covers key concepts such as complex numbers, analytic functions, cauchy's integral theorem, laurent series, and residue theorem. Numerous examples and exercises to illustrate the concepts and aid in understanding. It is suitable for students studying complex analysis in mathematics or related fields.
Typology: Schemes and Mind Maps
1 / 18
This page cannot be seen from the preview
Don't miss anything!
Complex
of the (^) quadratie (^) equotion,
Iathere
part
Im 2 y
The
,
-B ± CB-4ac)
2A
this lead: to ch velopmtnt of compla
variablks
2A
rsesented
heal part
ReRe ZZ^2
zo implies y-o
Varioble
J-)
: imginng soluhi,
as
algebsoit pera tions os tso Comple
inginay
(z+y,) +(ant)
(zj2-) (z1ti)^ (^
ti)
Z
21 t
L
( 2 Assoative^ law of^ additon
(io Co mnutatin
cilg Asso^ ciatve^ lacs^ of^
mu ttiplicatio,
Z, (2,2)
(èu) Dstributive^ la
las of^ multh^ icg^ ton
Z (2t 2)
(2,2) 2
r2r2>)* =2,* 22"
2
Tmportant Re latlas 8 CorÊeib4-io
2
Cos(io) = co b()
t, e
the product^ of^ two^ Compl^
wriablu,
-Por the^ case^ of^ n'^
terms.
When 2,^21 -
Cos
: n,^ h^ cos^ (9,^ to)^ +^ istà^ (9, +0))
nth noot^ of^ cony^ lex
2
2
no.
vGra
The nth Toot^ of^ Z^ mas^ be^ wsiten^ as
for disfin ct root ro! 2
-IP f(z) is onalyie
do
Consttutes the
all
sufticient condi^ tion^ )
Amalytie
froc tion at
tHarmonia functlón s
TF van d
’ No t analytE
Ca uch4 - Riemonn cg
conti'n dous
to
)
second order, then
in aginay pont Case I -
) Find portial d rivatives,
Case -
oy
V:e
and
Va (20)
can fint fl)
ls
Real or
74 2e?
aY_vesioytalsin)- (ycosy tsiay))
e (-^ x^ slh^ y-ycory^ -cing)
and finishig
the repuon
at
Proof -
Conne cfed rgon
J
rfone 9aid^ to^ be^ multiyl;
ano tlies (^) potst , 2 , (^) searctu
(a) slmplysimply^ connecfed
Acc. to stokes
C
Fiut te m
(6) mu^ lhply^ Conne tfed
theosen
fwo dnostons^ i^
becomy
Se con cl
H the
Cauchy
fy
statistrd
Ad
dsfvox t udy -)
2 20
According to^ thic^ Ah^ vla
27f o)
(4)
Lo pli) d
C
Taylor's^ Theore^ mt
let
2-
and ra diu R,
Dro a
cce C,^ wih5witth^ itsits^ cece^ nthenthe^ at^ the^ poin
M.
then at each Bt. 2
(Ar
n
(2-o) la).
Consi des ang^ porh^ t^ z^ in^ side
2/
pnra) (z-a) +..
SY) C.
a-(7-a)
Zo riside
Zo ouri'de
cê cle C, enclosing that point ?uth abe a
C
As
s
APpltng 8inonial
th
by-a
Strres
theorem
|+ 2-9 +
(h-a)' (w- a)
Multiply ) by fltA)
+(2-) -2-) (wea)nl
conVrf a^ uniformly,^ #rrce^ the^
sers
+(-0°(Ja
) l2-/
(wa
8y int oducin a
Uring Cauchy's Cntqra Pormula
CI
Integral along
neqative
Taylor': Serru
w. 2
a cros^ Cut^ AB,^ mutt-^ connected
tonVehkd a sinply^ connec^ lec
27 -
Tote gratr
27,
-a
+(2-) lus
alon
For fås^ t^ int^ gral^ Aw)^ expanded^ ex^ etty u'ke
271
clock cs^ (se.^ soitis
lies On C.
C
ABand BA^ cnce^ e
|w- al
In th seconcd intqral, z lies
So hete
Multipy by flw,^ we
w-
Thte qratng
2/ C
w. 2
Cu-Qta-
2-a
27i
2 -a
2-a
e-a)
(w-a)-()
2
(?-a)(
lw-) f(wt 2Ai (2-c)
2
lw. 2iC2-a)
b (2-a) 3
(2-a) 21 (-a-!
olwt
here L :
