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


z

nz

e

...n!

z...

!

z

!

zze z

iy xz

of seriesMaclaurin' called is

then Let

321)(exp

,)1(32

21

2

121

21 ,. zzz

zzz e

e

eeee

zz


,sin,cos

)sin(cos

)()2(

yevyeu

ivu

yi ye

e.e

eezf

xx

x

iyx

iyxz

Let


xyyx

xy

xx

xy

xx

vuvu

yevyev

yeuyeu

,

cos,sin

sin,cos

continuous are clearly

and satisfied are equations-CR Thus

yxyx v,v,,uu


zz

ziyxxx

xx

eedz

d

eeeyeiye

vi u zf

zf

.sincos

)(

)( and abledifferenti is


,.3 iyxz eee )( yiyeiy sincos

1sincos 22 yyeiy

Rxeeee xxxz 0 as

. numbercomplex any for zez 0


yee

eeeezx

iiyxz

&0

,.

when

may write We

...2,1,0

,2arg

n

nyez


12sin2cos

02sin12cos)4(2

iπe

π & π πi Hence

01

1sincos

i

i

e

ie

1sincos ie i


iie i 2/sin2/cos2/

i

ie i

2/sin2/cos2/


ziziz eeee 22 .).5(

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

.22

nee

i

e

zinz

z

period

imaginarywithperiodicis


Cze

xez

x

ifnegativebemay But

0).6(

1zez that such Find :Example

iiyx

z

eee

e.1.

1

:Solution


nyx

nny

ex

2&0

...2,1,0,2

,1

and

1

210

).12(

ze

,...,, n

, in

iy xz

then

if Thus,


anywhere.analytic not is

:Excerciseze)7(

ie

z z 112

that such of values all Find Q.


then, real is If (1) x

,2

cosixix ee

x

.2

sini

eex

ixix


define wecomplex, is ifSimilarly z

,2

cosiziz ee

z

)1(2

sin

i

eez

iziz

,sincos zizeiz


zzec

zz

z

zz

z

zz

sin

1cos,

cos

1sec

,sin

coscot,

cos

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()(

zzdz

diii cossin)(


zzdz

div sincos)(

zzdz

dv 2sectan)(


2121

21

sincoscossin

)sin()(

zzzz

zzvi

2121

21

sinsincos.cos

cos)(

zzzz

zzvii


then Put

,0

.2

sin

,2

cos4

x

i

eez

eez

iziz

iziz


2cos

iyiiyi eeiy

yee yy

cosh2


)(2

1sin yy ee

iiy

yi

eei yy

sinh

)(2

1


)cos(cos iyxz

)(sin.sin)(coscos iyxiyx

hyxiyx sin.sincosh.cos


hyxiyx

iyxiyx

iyxz

sin.coscosh.sin

sin.coscos.sin

sinsin


yhxz

yx z

222

222

sincoscos

sinhsinsin

(Exercise) Hence

1sincos

1sinhcos22

22

xx

xxh

(Use) :Hints


zz,

z

z z

z z &

cos

1sec

cos

sintan

sectan

: oficity (5).Analyt

0cos

sec&tan

z

zz

where

points the atexcept everywhere

analytic are


0sinsincoscos

cos

0cos

hyxihyx

iyx

z

0sinhsin

&,0coscos

yx

hyx


0cosh y

)010

1

2

1

2cosh(

2

y

yy

yy

e

ee

eey

...2,1,0,2

120cos nnxx


2

120sin

nxx for But

0102

sinh

00sinh

2 yeee

y

yy

yyy


,...21,02

12

sectan2

12

n,π

nz

z z &

niyxz

atexcept eevery wher

analytic are


0sin

cos&cot

sin

1cos

sin

coscot

coscot)(

z

zecz

zzec&

z

z z

ec z: z &

where

points the at except everywhere

analytic are

ofty Analytici6Ex.


.2

cosh

,2

sinh

zz

zz

eez

eez

:Definition


.everywhere

analytic are

everywhere

analytic are

zz

ee zz

cosh&sinh

&).1(


zee

ee

dz

dz

dz

d

zz

zz

cosh2

2sinh).2(

zzdz

dsinhcosh Similarly,


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

1sinhcosh 22 zz

zz coshcosh


),cosh(cos).4( izz

zee

zi

eez

zizi

zz

cos2

cosh

2cosh


zz

zizi

ziz

coshcosh

coshcos

coshcos

2

zzi coshcos).5(


ziiz sinhsin).6(

zi

zizi.

sinh

sinhsin)7(


2121

21

sinh.coshcosh.sinh

sinh).8(

zzzz

zz

2121

21

sinh.sinhcosh.cosh

cosh).9(

zzzz

zz


:Soln

yxiyx

z

sin.coshcos.sinh

sinh.10

zizi sinhsin


)sin(

)sin(sinh

yixi

iziz

]sin)cos(

cos)[sin(

yix

yixi


]sincosh

cossinh[

yx

yxii

yxiyx

z

sincoshcossinh

sinh


yxz 222sinsinhsinh

: Excercise


yxi

yxza

sin.sinh

cos.coshcosh)(

Similarly

)cos(coscosh yixziz Use

yxzb 222cossinhcosh)(


:sec&tanh).11( hzzofyAnalyticit

.cosh

1sec

,cosh

sinhtanh

zhz

z

zz


.0cosh

sec&tanh

z

hzz

wherepoints

the at except everywhere

analytic are


0cosh zNow

0coscos yixzi

0sin.sincos.cos yixyxi

0sin.sinhcos.cosh yxiyx


.0sin.sinh

,0coscosh

yx

yx.

and

,...2,1,0,2

12

0cos0cosh

nny

yx


00sinh

0sin,2

12

xx

yny

For

...,2,1,0

,2

12

n

in

iyxz


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

12

sec&tanh

ni

nz

hzz

at except

everywhere analytic

are


Exercise:

coth z and cosech z are analytic

everywhere except at z = ni,

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


zzzii

zzzi

ImcoshcosImsinh)(

ImcoshsinImsinh)(

:

thatShowQ.

yhxi

yxzi

sin.cos

cosh.sinsin)(

:Sol


yx

yxz22

222

sinh.cos

cosh.sinsin

yx

yx22

22

sinh.sin1

sinh1sin

yx 22 sinhsin


y

y

yxzy

2

2

2222

cosh

sinh1

sinhsinsinsinh

yzy coshsinsinh


yx

yxz

yxi

yxzii

22

222

sinh.sin

cosh.coscos

sinh.sin

cosh.coscos)(


yx

yx

yxz

22

22

222

sinhcos

sinh.cos1

sinh1coscos


hyzyh

y

yh

yhxzyh

coscossin

cosh

sin1

sincoscossin

2

2

2222


z, w

z,

iy xz

log

log

i.e.

by denoted is

of logarithm natural The

for defined is and 0log z z

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

relation theby


zw

zew

log

,

writewethen ifi.e.

zArg

er

ririyxz

ivuw

i

,

,

sincos

,

where

Let


iivu

iivu

eree

erei

.

)(Then

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

,2

,

n

nv

zreu


nv

zru

2

,lnln

niz

viuzw

2ln

log


integerany is

and

Since

n

nz

zArg

,2arg

,

0,arglnlog zzizz


Arg z z n arg0 then , When

.0ln

log

log0

zz,Argiz Log z

Log z

z

zn

i.e. ,by denoted is and

of value principal the called is

of value the then , When


niiz

niz

zizz

2ln

2ln

arglnlog

,...2,1,0

,2log

n

nizLogz


,.....2,1,0

,2ln

arglnlog

n

niz

ziz z

Since

1:Remark

function. dmultivaluea is zlog


zArg

izzLog

Since

:2 Remark

,ln

function.

valued-single a is zLog


.00

ln2

1ln 22

,

yxz

at except

everywhere continuous is

:3Remark


2,0

,ln

lnlog

r

ir

izzzf

consider and

number, realany be Let :4 Remark

,,ln, rvrru


y

x

}20{

log

,zz:D

z

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)(

1

zrzzD

θ i r z i

domain in

analytic is

:6 Remark

})(,0:{

ln)(

2

zArgrzzD

irLog z ii

domain

the inanalytic is


1,0

0,1

,,ln,,

vv

ur

u

rvrru

r

r

As

.continuous are sderivative

partial order-first and satisfied are

form polar in equations-CR

rθθr vru,vur


rri viuez

dz

dzf log

1

11D

zer iin

.1

2Dz

zLogdz

din

when,particular In


axis. realnegative oni.e.,-

ray the on and at except plane

complex wholethe onanalytic is

7:Remark

00,

Log z

.and

by given are of essingularti i.e.

0Im0Re z z

Log z


.

)(

zfz D, Fz

SD

zF

Szf

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


ziz z

z

θizz

arglnlog

2,0

,lnlog

of branch a is

, fixed each For Ex.


branch. principal the called is

,0

,ln

z

izzLog


.01

x y

izLog

line halfthe on except

everywhereanalytic is

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

bygivenis ofy singularit

have We:Solution

zf

izLogzf


1&0

01Im

&01Re

0Im&0Re

yx

yix

yix

iziz

y=1 O


axis. real the of portion

the on and points the

at except everywhereanalytic is

functionthe thatShow b9Q

4

21

42

x

/i

iz

zLogzf


thennumbers,complex

zeronontwoanybe If 21 z&z

2121 logloglog)1( zzzz

212

1 logloglog)2( zzz

z


2121 zLogzLogzzLog

But

212

1 zLogzLogz

zLog

zLognzLog n


1,1 21 zzLet(1)Ex

z1 z2 z1

z1= -1+i0


111 ln zArgizzLog

i

zArgiLog

0

1ln1 1

izLogzLog 221


00.0

ln

1.

212121

21

i

zzArgizzzzLog

zz

But

2121 zLogzLogzzLog

Thus


iLogiLog 121 297 p. Q.3(b)

iLog

iiLog

iLog

2

21

12

2

LHS

iiArgi 22ln


22ln

22ln

i

i


iLog 12RHS

iArgii 11ln2

4

32ln2

i


RHSLHS

2

32ln

]4

32ln

2

1[2

i

i


4

9

4,0

,lnlog

,log2log)( 2

zr

irz

iia

when

thatShow:p.97Q.4


4

11

4

3

,0

,lnlog

,log2log 2

when

zr

irz

iib


zcc

c

ez

iszc

z

log

0)1(

as

defined no.Thencomplex any is

and no.,complex a be Let

.

log

c

Logzcc

z

ez

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

inni

2

122

20


is of value Principal

real, is whichi

ni

i

e i ,2

12

).0(2

ne


. of P.V. Findb i-i

2

2

12

2

12

log

,..2,1,0,

ei

ne

eei

i

n

iniiii

of value Principal

:Solution


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

iπ

iLog 2

have We

iiiiiLog

2arg

2ln

2loglog

ni 2

22ln

2

2ln

iiLogLog is

of value Principal


ii

11 of value principal the FindQ.

- / 4

1

-i

iii ei 1log111:Solution


102

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Lecture 27 29