Lecture 27 29

BITS Pilani Pilani Campus

MATH F112 (Mathematics-II)

Complex Analysis

BITS Pilani Pilani Campus

Lecture 27-30 Elementary Functions

Dr Trilok Mathur, Assistant Professor, Department of Mathematics

BITS Pilani, Pilani Campus

1. Exponential Functions

2. Trigonometric Functions

3. Hyperbolic Functions

4. Logarithmic Functions

5. Complex Exponents

Self Study (Sec 36, p.112-115)

6. Inverse Trigonometric Functions

7. Inverse Hyperbolic Functions

of seriesMaclaurin' called is

then Let

321)(exp

,)1(32

21 ,. zzz

,sin,cos

)sin(cos

yevyeu

yevyev

yeuyeu

cos,sin

sin,cos

continuous are clearly

and satisfied are equations-CR Thus

yxyx v,v,,uu

ziyxxx

eeeyeiye

vi u zf

.sincos

)( and abledifferenti is

,.3 iyxz eee )( yiyeiy sincos

1sincos 22 yyeiy

Rxeeee xxxz 0 as

. numbercomplex any for zez 0

eeeezx

may write We

...2,1,0

12sin2cos

02sin12cos)4(2

π & π πi Hence

1sincos

1sincos ie i

iie i 2/sin2/cos2/

2/sin2/cos2/

ziziz eeee 22 .).5(

,...3,2,1,0,

period

imaginarywithperiodicis

ifnegativebemay But

1zez that such Find :Example

:Solution

...2,1,0,2

,...,, n

if Thus,

anywhere.analytic not is

:Excerciseze)7(

z z 112

that such of values all Find Q.

then, real is If (1) x

cosixix ee

define wecomplex, is ifSimilarly z

cosiziz ee

,sincos zizeiz

coscot,

sintan

(2). Since ez is analytic z and linear

combination of two analytic functions is

again analytic, hence it follows that

sin z and cos z are analytic functions.

:prove toeasy is it Using )1()3( .

zzi sin)sin()(

zzii cos)cos()(

diii cossin)(

div sincos)(

dv 2sectan)(

sincoscossin

)sin()(

sinsincos.cos

then Put

iyiiyi eeiy

yee yy

1sin yy ee

eei yy

)cos(cos iyxz

)(sin.sin)(coscos iyxiyx

hyxiyx sin.sincosh.cos

hyxiyx

iyxiyx

sin.coscosh.sin

sin.coscos.sin

sinsin

sincoscos

sinhsinsin

(Exercise) Hence

1sincos

1sinhcos22

(Use) :Hints

sintan

sectan

: oficity (5).Analyt

sec&tan

points the atexcept everywhere

analytic are

0sinsincoscos

hyxihyx

0sinhsin

&,0coscos

0cosh y

2cosh(

...2,1,0,2

120cos nnxx

120sin

nxx for But

00sinh

2 yeee

,...21,02

sectan2

atexcept eevery wher

analytic are

cos&cot

coscot

coscot)(

ec z: z &

points the at except everywhere

analytic are

ofty Analytici6Ex.

:Definition

.everywhere

analytic are

everywhere

analytic are

cosh&sinh

2sinh).2(

dsinhcosh Similarly,

z--z. sinh)sinh()3(

1sinhcosh 22 zz

zz coshcosh

),cosh(cos).4( izz

coshcosh

coshcos

zzi coshcos).5(

ziiz sinhsin).6(

sinhsin)7(

sinh.coshcosh.sinh

sinh).8(

sinh.sinhcosh.cosh

cosh).9(

sin.coshcos.sinh

sinh.10

zizi sinhsin

)sin(sinh

]sin)cos(

cos)[sin(

]sincosh

cossinh[

sincoshcossinh

yxz 222sinsinhsinh

: Excercise

sin.sinh

cos.coshcosh)(

Similarly

)cos(coscosh yixziz Use

yxzb 222cossinhcosh)(

:sec&tanh).11( hzzofyAnalyticit

sinhtanh

.0cosh

sec&tanh

wherepoints

the at except everywhere

analytic are

0cosh zNow

0coscos yixzi

0sin.sincos.cos yixyxi

0sin.sinhcos.cosh yxiyx

.0sin.sinh

,0coscosh

,...2,1,0,2

0cos0cosh

00sinh

0sin,2

...,2,1,0

,....2,1,0,2

sec&tanh

at except

everywhere analytic

Exercise:

coth z and cosech z are analytic

everywhere except at z = ni,

,.....2,1,0 n

ImcoshcosImsinh)(

ImcoshsinImsinh)(

thatShowQ.

sin.cos

cosh.sinsin)(

sinh.cos

cosh.sinsin

sinh.sin1

sinh1sin

yx 22 sinhsin

sinhsinsinsinh

yzy coshsinsinh

sinh.sin

cosh.coscos

sinh.sin

cosh.coscos)(

sinhcos

sinh.cos1

sinh1coscos

yhxzyh

coscossin

sincoscossin

by denoted is

of logarithm natural The

for defined is and 0log z z

)....(.......... izew

relation theby

writewethen ifi.e.

ririyxz

sincos

)(Then

,......2,1,0

integerany is

0,arglnlog zzizz

Arg z z n arg0 then , When

zz,Argiz Log z

i.e. ,by denoted is and

of value principal the called is

of value the then , When

arglnlog

,...2,1,0

nizLogz

,.....2,1,0

arglnlog

1:Remark

function. dmultivaluea is zlog

izzLog

:2 Remark

function.

valued-single a is zLog

1ln 22

at except

everywhere continuous is

:3Remark

consider and

number, realany be Let :4 Remark

,,ln, rvrru

domain the in continuous

and valued- single is Then

Remark 5: The function log z is NOT continuous on the line θ = α as arg z is NOT continuous on the line θ = α .

For if z is a point on the ray = then there are points arbitrary close to z at which the values of v are nearer to , and also there are points such that the values of v are nearer to + 2.

exist.notdoes zαz

arglim

}2)arg(,0:{

lnlog)(

θ i r z i

domain in

analytic is

:6 Remark

})(,0:{

zArgrzzD

irLog z ii

domain

the inanalytic is

,,ln,,

.continuous are sderivative

partial order-first and satisfied are

form polar in equations-CR

rθθr vru,vur

rri viuez

dzf log

zer iin

zLogdz

when,particular In

axis. realnegative oni.e.,-

ray the on and at except plane

complex wholethe onanalytic is

7:Remark

by given are of essingularti i.e.

0Im0Re z z

zfz D, Fz

of values the of one is

all for that such domain some

inanalytic isthat function valued

singleany is set a on defined

function valued-multiple a of branchA

:Definition

arglnlog

,lnlog

of branch a is

, fixed each For Ex.

branch. principal the called is

izzLog

line halfthe on except

everywhereanalytic is

function the thatShow :97 p. Q.9(a)

bygivenis ofy singularit

have We:Solution

izLogzf

0Im&0Re

axis. real the of portion

the on and points the

at except everywhereanalytic is

functionthe thatShow b9Q

zLogzf

thennumbers,complex

zeronontwoanybe If 21 z&z

2121 logloglog)1( zzzz

1 logloglog)2( zzz

2121 zLogzLogzzLog

1 zLogzLogz

zLognzLog n

1,1 21 zzLet(1)Ex

z1 z2 z1

z1= -1+i0

111 ln zArgizzLog

zArgiLog

1ln1 1

izLogzLog 221

212121

zzArgizzzzLog

2121 zLogzLogzzLog

iLogiLog 121 297 p. Q.3(b)

iiArgi 22ln

iLog 12RHS

iArgii 11ln2

RHSLHS

,lnlog

,log2log)( 2

thatShow:p.97Q.4

,lnlog

,log2log 2

defined no.Thencomplex any is

and no.,complex a be Let

Logzcc

Log z,z

of value principal the calledis

then by replaced is If

value. principal its find

and real is thatShow :p.104 aQ.2 ii

iiii arglnlog

iii ei log :Soln

is of value Principal

real, is whichi

e i ,2

. of P.V. Findb i-i

,..2,1,0,

iniiii

of value Principal

:Solution

iba iLog of terms in Write(c) )log(

iLog 2

have We

iiiiiLog

2loglog

iiLogLog is

of value Principal

11 of value principal the FindQ.

iii ei 1log111:Solution

Lecture 27 29

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