1

Chapter 3. Elementary Functions

Consider elementary functions studied in calculus and define corresponding functions of a complex variable.

To be specific, define analytic functions of a complex variable z that reduce to the elementary functions in calculus when z = x+i0 .

23. Exponential Function

If f (z), is to reduce to when z=x

i.e. for all real x, (1)

It is natural to impose the following conditions:

f is entire and for all z. (2)

As shown in Ex.1 of sec.18

is differentiable everywhere in the complex

plane and .

z x iy xe

( 0) xf x i e

'( ) ( )f z f z

( ) (cos sin )xf z e y i y '( ) ( )f z f z

2

It can be shown that (Ex.15) this is the only function satisfying conditions

(1) and (2).

And we write

(3)when Euler’s Formula

( ) expzf z e z (cos sin )xe y i y

, cos siniyz iy e y i y

, ,

z x iy

z i x

e e e

if e e then e y

(5)

since is positive for all x

arg( ) 2 ( 0, 1, 2,...)

0

x

z x

z

z

z

e

e e

e y n n

e

e

and

since is always positive,

for any complex number z.

3

• can be used to verify the additive property

1 2 1 2(exp )(exp ) exp( )

z x iye e e

z z z z

1 1 2 2 1 2 1 2

1 2 1 2

1 1 2 2 1 2

1 1 1 2 2 2

1 2

( ) ( )

( ) ( )

,

(exp )(exp ) ( )( )x iy x iy x x iy iy

x x i y y

x iy x iy z z

z x iy z x iy

z z e e e e e e e e

e e

e e

1 2 2 1

11 2

2

exp( )exp( ) exp

expexp( )

exp

z z z z

zz z

z

0 1Since 1, .z

ze e

e

4

Ex : There are values of z such that 1ze

1 1, 2 ( 0, 1, 2, )

0, (2 1) ( 0, 1, 2, )

x iy i xe e e e y n n

x z n i n

2 2

(exp ) exp( ) 0, 1, 2,...

since

is periodic with a pure imaginary period 2 .

n

z i z i z

z

z nz n

e e e e

e i

5

24. Trigonometric Functions

By Euler’s formula

It is natural to define

These two functions are entire since are entire.

cos sin

sin2

ix

ix ix

e x i x

e ex

i

cos sin

cos2

ix

ix ix

e x i x

e ex

sin cos2 2

iz iz iz ize e e ez z

i

,iz ize e

sin cos2 2

cos sin2 2

iz iz iz iz

iz iz iz iz

d ie ie e ez z

dz i

d ie ie e ez z

dz i

also can obtain

sin( ) sin , cos( ) cos .z z z z

6

Ex:

1 1 2 2

1 2 1 2 1 2 1 2

1 2 1 2 1 2

1 2

1 2 1 2

2sin cos sin( ) sin( )

2sin cos 2( )( )2 2

2 2sin( ) sin( )

iz iz iz iz

i z z i z z i z z i z z

z z z z z z

e e e ez z

i

e e e e

i iz z z z

1 2 1 2 1 2

1 2 1 2 1 2

2 2

2 2

sin( ) sin cos cos sin (5)

cos( ) cos cos sin sin (6)

sin cos 1

sin 2 2sin cos

cos 2 cos sin

z z z z z z

z z z z z z

z z

z z z

z z z

sin( ) cos2

sin( ) cos2

z z

z z

7

when y is real.sinh

2

sin2

y y

iy iy

e ey

e ey

i

cosh2

cos2

y y

iy iy

e ey

e ey

1 2

( )sin( ) sinh

2 2

cos( ) cosh2

, in (5) and (6)

sin( ) sin cos cos sin

sin sin cosh cos sinh (11)

cos( ) cos cos sin sin

cos cos cosh sin sinh (12

y y y y

y y

e e i e eiy i y

i

e eiy y

let z x z iy

x iy x iy x iy

z x y i x y

x iy x iy x iy

z x y i x y

)

2 2 2

2 2 2

sin sin sinh (15)

cos cos sinh (16)

z x y

z x y

• in Exercise 7.

unbounded

8

• A zero of a given function f (z) is a number z0 such that f (z0)=0

Since

And there are no other zeros since from (15)

sin sin when

sin 0, when ( 0, 1, 2,...)

real

z x z x

z z n n

2 2sin sinh 0

sin 0 sinh 0

0

x y

x and y

x n y

sintan

coscos

cotsin

1sec

cos1

cscsin

zz

zz

zz

zz

zz

2

2

tan sec

cot csc

sec sec tan

csc csc cot

dz z

dzd

z zdzd

z z zdzd

z z zdz

9

sinh2

cosh2

sinh cosh2

cosh sinh

z z

z z

z z

e ez

e ez are entire

d e ez z

dzd

z zdz

25. Hyperbolic Functions

since sin , cos2 2

sinh( ) sin

cosh( ) cos

sin( ) sinh

cos( ) cosh

iz iz iz ize e e ez z

ii iz z

iz z

i iz z

iz z

(3)

(4)

10

Frequently used identities

2 2

1 2 1 2 1 2

1 2 1 2 1 2

2 2 2

2

sinh( ) sinh

cosh( ) cosh

cosh sinh 1

sinh( ) sinh cosh cosh sinh

cosh( ) cosh cosh sinh sinh

sinh sinh cos cosh sin

cosh cosh cos sinh sin

sinh sinh sin

cosh sin

z z

z z

z z

z z z z z z

z z z z z z

z x y i x y

z x y i x y

z x y

z

2 2h cosx y

11

•

•

•

sinhtanh

cosh

zz

z

2 2

2

tanh sech , coth csch

sech sech tanh , csch csch coth

d dz z z z

dz dzd d

z z z z z zdz dz

• From (4), sinhz and coshz are periodic with period •

•

2 isinh 0 ( 0 1, 2,...)

cosh 0 ( 0 1, 2,...)2

z iff z n i n

z iff z n i n

12

26. The Logarithmic Function and Its Branches

To solve . 0

, ( )

, 2 ( integer).

ln ( 2 ) ( 0, 1, 2,...).

w

i

u iv i

u

e z for w z

let w u iv when z re

e e re

e r v n n

w r i n n

Thus if we write

log

log ln ( 2 ) ( 0, 1, 2,...) (2)

we get (3)z

z r i n n

ze

13

Now,• If z is a non-zero complex number, , then is any of

, when

iz re 2 ( 0, 1, 2,...)n n Argz

log ln (4)

log ln arg ( 0) (5)

z r i

or z z i z z

• Note that it is not always true that

since has many values for a given z or ,

log

log

z

z

e z

e

, arg( ) 2 ( 0, 1, 2,...)z x ze e e y n n

From (5), log( ) ln arg( )

( 2 )

log 2 ( 0, 1, 2,...)

z z z

z

e e i e

x i y n

e z n i n

14

Log ln

Log ln Arg ( 0)

log Log 2 ( 0, 1, 2,...)

Ex. log1 2

log( 1) (2 1)

z r i

or z z i z Z

z z n i n

n i

n i

The principal value of log z is obtained from (2) when n=0 and is denoted by

Log . z

15

• If we let denote any real number and restrict the values of in expression (4) to the interval then

2

log ln ( 0, 2 ) (9)

( , ) ln

( , )

z r i r

u r r

v r

with components

is single-valued and continuous in the domain.

is also analytic,log (9)z in1 1

,

1, 0, 0, 1

r r

r r

u v u vr r

u u v vr

1 1log ( ) ( 0)

1log ( 0, arg 2 )

i ir r i

dz e u iv e i

dz r red

z z zdz z

16

A branch of a multiple-valued function is any single-valued

function that is analytic in some domain at each point of

which the value ( ) is one of the values ( ).

For each fixed t

f

F z

F z f z

he single-valued function (9) is a branch

of the multiple valued function (4).

The function Log = log ( 0, ) is called

the principal branch.

A branch cut is a portion of a line or curv

z r r

e that is introduced

in order to define a branch of a multiple-valued function .

Points on the branch cut for are singular points of , and

any point that is common to all branch cuts of

F f

F F

f

is called a

branch point.

17

The origin and the ray = make up the branch cut for the

branch (9) of the logarithmic function.

The branch cut of the principal branch (13) consists of the

origin and the ray = .

The origin

is a branch point for branches of the multiple-valued

logarithmic function.

18

27. Some Identities Involving Logarithms

non-zero. complex numbers1 2,z z

1 2 1 2log( ) log logz z z z (1)Pf:

1 2 1 2

1 2 1 2

1 2 1 2

ln ln ln

arg( ) log log (sec.6)

z z z z

z z z z

also z z z z

1 2 1 2 1 1 2 2

1 2 1 2

11 2

2

ln arg( ) ln arg ln arg

sec.26.(5) log( ) log log

log( ) log log

z z i z z z i z z i z

from z z z z

zsimilaiy z z

z

19

Example: (A)

(B)

also

1 2

1 2

1 2

1 2

1

1

log , log

log( ) 0

z z

z z

if z i z i

z z

Then (1) is satisfied when is chosen.

1 2 1 2

1 2 1 2

Log( ) 0, Log Log 2

Log( ) Log Log

z z z z i

z z z z

log

1

( 0, 1, 2,..)

1exp( log ) ( 1,2,..)

n n z

n

z e n

z z nn

has n distinct values which are nth routs of z

Pf: Let exp( ), arg

1 1exp( log ) exp[ (ln ( 2 )]

1 ( 2 )exp[ ln ]

2exp[ ] ( 0, 1, 2,...)n

z r i z

z r i kn n

i kr

n nk

r i kn n

20

28. Complex Exponents

when , c is any complex number,

is defined by

where log z donates the multiple-valued log function.

( is already known to be valid when c=n and c=1/n )

Example 1: Powers of z are in general multi-valued.

0z cz log (1)c c zz e

cz cz

2 1exp( 2 log ) exp[ 2 (2 ) ]

2exp[(4 1) ] 0, 1, 2,...

ii i i i n i

n n

since

loglog

2

1

1 1

1exp[(4 1) ] 0, 1, 2,...

zz

c z cc c z

i

ee

e zz e

n ni

21

If and is any real number, the branch

of the log function is single-valued and analytic in the indicated domain.

when that branch is used,

is singled-valued and analytic in the same domain.

iz re

log ln ( 0, 2 )z r i r

exp( log )cz c z

1

Log

(exp( log )) [exp( log )]

exp( log )exp( 1 log )

exp(log )

The principal value of occurs when log z is replaced by Log z

in (1)

c

c

c

c zc

d d cz c z c z

dz dz zc z

c c c z czz

z

z e

Example 2. The principal value of ( - ) is

exp[ Log( )] exp[ ( )] exp2 2

ii

i i i i

22

0,r

In (1)log log( )

cc c z zz e e

log log( )zz z c cc e e

now define the exponential function with base C.

when a value of logc is specified, is an entire function of z. z ze czclog log log

log

z z c z c

z z

d dc e e c

dz dzdc c c

dz

3 22 2 2 2exp[ Log ] exp( ln ) exp( )

3 3 3 3z r i r i

Example 3. The principal value of2

3z

It is analytic in the domain

23

29. Inverse Trigonometric and Hyperbolic Functions

write1

2

sin ( sin )

when2

( ) 2 ( ) 1 0

iw iw

iw iw

w z z w

e ez

i

e iz e

iweSolving for

taking log on both sides.

122(1 ) (1)iwe iz z

12

12

12

2

2

1 2

log[ (1 ) ]

log[ (1 ) ]

sin log[ (1 ) ]

iw iz z

w i iz z

or z i iz z

24

Example:

But

since

1

1 1

sin ( ) log(1 2)

log(1 2) ln(1 2) 2 ( 0, 1, 2,...)

log(1 2) ln( 2 1) 2 1 ( 0, 1, 2,...)

( 2 1)( 2 1) 1ln( 2 1) ln ln ln(1 2)

( 2 1) 1 2

log(1 2) ( 1) ln(1 2) 0, 1, 2,...

sin ( ) ( 1) ln(1

n

n

i i

n i n

n i n

n i n

i n i

2) ( 0, 1, 2,...)n

similarly, 1

21 2

1

cos log[ (1 ) ]

tan log2

z i z i z

i i zz

i z

25

12

12

12

12

1 12 2

1 1 12 2 2

12

211 2

2

2

2

2 2

2 2 2

2

(1 ) ( 2 )sin

(1 )

(1 )

(1 )

1 (1 ) (1 ) 1

(1 ) (1 ) (1 )

1

(1 )

i z zdz i

dz iz z

i z ziiz z

iz z z iz

iz z iz z z

z

12

1

2

12

1cos

(1 )

1tan

1

dz

dz z

dz

dz z

12

12

1 2

1 2

1

sinh log[ ( 1) ]

cosh log[ ( 1) ]

1 1tanh log

2 1

z z z

z z z

zz

z