Chapter 3. Elementary Functions

Weiqi Luo (骆伟祺 )School of Software

Sun Yat-Sen UniversityEmail ： weiqi.luo@yahoo.com Office ： # A313

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The Exponential Functions The Logarithmic Function Branches and Derivatives of Logarithms Some Identities Involving Logarithms Complex Exponents Trigonometric Function Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions

2

Chapter 3: Elementary Functions

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The Exponential Function

29. The Exponential Function

3

,z x iye e e z x iy

cos siniye y i y

According to the Euler’ Formula

cos sinz x xe e y ie y

Note that here when x=1/n (n=2,3…) & y=0, e1/n denotes the positive nth root of e.

u(x,y) v(x,y)

Single-Valued

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Properties

29. The Exponential Function

4

1 2 1 2z z z ze e e

1 1 1 2 2 2; +iyz x iy z x Let

1 1 2 2 1 1 2 2x +iy x +iy x iy x iy( e )( e )e e e e1 2 1 2x x iy iy( )(e e )e e

1 2 1 2x x x x=ee e Real value:

Refer to pp. 18

1 2 1 2iy iy i(y )e e e y1 2 1 2x x i(y )e e y

1 2 1 2 1 2( )+ ( y )z z x x i y 1 2z +ze

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Properties

29. The Exponential Function

5

1

1 2

2

zz z

z

ee

e1 2 2 1z z z ze e e

Refer to Example 1 in Sec 22, (pp.68), we have that

z zde e

dz everywhere in the z plane

which means that the function ez is entire.

2 0ze

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Properties

29. The Exponential Function

6

0ze For any complex number z

z x iy ie e e re &xr e y

| | 0 & arg( ) 2 ( 0, 1, 2,...)z x zr e e e y n n

2 2z i z ie e e 2 2, cos 2 sin 2 1z i z ie e e i

which means that the function ez is periodic, with a pure imaginary period of 2πi

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Properties

29. The Exponential Function

7

0xe For any real value x

while ez can be a negative value, for instance

cos sin 1ie i

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Example

In order to find numbers z=x+iy such that

29. The Exponential Function

8

1ze i /42z x iy ie e e e

/42 &x iy ie e e 1

ln 2 & 2 ,( 0, 1, 2,...)2 4

x y n n

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

2 4z i n n

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pp. 92-93

Ex. 1, Ex. 6, Ex. 8

29. Homework

9

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The Logarithmic Function

30. The Logarithmic Function

10

log ln ( 2 ), ( 0, 1, 2,...)z r i n n 0iz re

It is easy to verify that

log ln ( 2 ) ln ( 2 )z r i n r i n ie e e e re z

Please note that the Logarithmic Function is the multiple-valued function.

iz re ln r i ln ( 2 )r i ln ( 2 )r i …

One to infinite values

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The Logarithmic Function

30. The Logarithmic Function

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ln | | arg( )z i z

log ln ( 2 ), ( 0, 1, 2,...)z r i n n 0iz re

Suppose that 𝝝 is the principal value of argz, i.e. -π < ≤𝝝 π

g ln ( ) lnLo z r iArg z r i is single valued.

And

log 2 , 0, 1, 2,...z Logz i n n

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Example 1

30. The Logarithmic Function

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log( 1 3 ) ?i

( 2 /3)log( 1 3 ) log(2 )ii e

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

3i n n

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Example 2 & 3

30. The Logarithmic Function

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log1 ln1 (0 2 ) 2 , 0, 1, 2,...i n n i n

1 0Log

log( 1) ln1 ( 2 ) (2 1) , 0, 1, 2,...i n n i n

( 1)Log i

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The Logarithm Function

where𝝝=Argz, is multiple-valued.

If we let θ is any one of the value in arg(z), and let α denote any real number and restrict the value of θ so that

The above function becomes single-valued.

With components

31. Branches and Derivatives of Logarithms

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log ln ( 2 ), 0, 1, 2,... z r i n n

log ln , ( 0, 2 )z r i r

2

( , ) ln & ( , )u r r v r

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The Logarithm Function

is not only continuous but also analytic throughout the domain

31. Branches and Derivatives of Logarithms

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log ln , ( 0, 2 )z r i r

0, 2r A connected open set

?

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The derivative of Logarithms

31. Branches and Derivatives of Logarithms

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log ln , ( 0, 2 )z r i r

( , ) ln & ( , )u r r v r

&r rru v u rv

1 1 1log ( ) ( 0)i i

r r i

dz e u iv e i

dz r re z

1og

dL z

dz z

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Examples When the principal branch is considered, then

31. Branches and Derivatives of Logarithms

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3( ) ( )Log i Log i

ln12 2i i

And

33 ( ) 3(ln1 )

2 2Log i i i

3( ) 3 ( )Log i Log i

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pp. 97-98

Ex. 1, Ex. 3, Ex. 4, Ex. 9, Ex. 10

31. Homework

18

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32. Some Identities Involving Logarithms

19

1 2 1 2log( ) log logz z z z

where 1 21 1 2 20 & 0i iz re z r e

1 21 2 1 2 1 2 1 2log( ) log( ) ln( ) ( 2 )i iz z re r e r r i n

1 2 1 1 2 2ln ln ( 2 ) ( 2 )r r i n i n

1 1 1 2 2 2[ln ( 2 )] [ln ( 2 )]r i n r i n

1 1 2 2(ln | | arg ) (ln | | arg )z i z z i z

1 2log logz z 1 2n n n 1 11

1 2 1 2 1 22

log( ) log( ) log log log logz

z z z z z zz

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Example

32. Some Identities Involving Logarithms

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1 2 1z z

1 2log( ) log(1) 2z z n i

1 2log( ) log( ) log( 1) (2 1)z z n i

1 2 1 2 1 2log log (2 1) (2 1) 2( 1)z z n i n i n n i

1 22 log( )n i z z 1 2 1n n n

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32. Some Identities Involving Logarithms

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log ( 0, 1, 2,...)n n zz e n

When z≠0, then

1log1/ ( 1,2,3...)zn nz e n

logc c zz e Where c is any complex number

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pp. 100

Ex. 1, Ex. 2, Ex. 3

32. Homework

22

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Complex Exponents

When z≠0 and the exponent c is any complex number, the function zc is defined by means of the equation

where logz denotes the multiple-valued logarithmic function. Thus, zc is also multiple-valued.

33. Complex Exponents

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logc c zz e

The principal value of zc is defined by

ogc cL zz e

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33. Complex Exponents

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iz re If and α is any real number, the branch

log lnz r i ( 0, 2 )r Of the logarithmic function is single-valued and analytic in the indicated domain. When the branch is used, it follows that the function

exp( log )cz c z

is single-valued and analytic in the same domain.

exp( log ) exp( log )cd d cz c z c z

dz dz z

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Example 1

33. Complex Exponents

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2 exp( 2 log )ii i i

1log ln1 ( 2 ) (2 ) , ( 0, 1, 2,...)

2 2i i n n i n

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

Note that i-2i are all real numbers

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Example 2

The principal value of (-i)i is

33. Complex Exponents

26

exp( ( )) exp( (ln1 )) exp2 2

iLog i i i

P.V. exp2

ii

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Example 3 The principal branch of z2/3 can be written

33. Complex Exponents

27

3 22 2 2 2exp( ) exp( ln ) exp( )

3 3 3 3Logz r i r i

Thus

23 32 23 2 2

cos sin3 3

z r i r

This function is analytic in the domain r>0, -π<𝝝<π

P.V.

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Example 4 Consider the nonzero complex numbers

33. Complex Exponents

28

1 2 31 , 1 & 1z i z i z i

2 ln 21 2( ) 2i i iLog iz z e e

When principal values are considered

(1 ) /4 (ln 2)/21i iLog i iz e e e

(1 ) /4 (ln 2)/22i iLog i iz e e e

-2 ln 22 3( ) ( 2)i i iLog iz z e e e （ ）

( 1 ) 3 /4 (ln 2)/23i iLog i iz e e e

1 2 1 2( )i i iz z z z

22 3 2 3( )i i iz z z z e

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The exponential function with base c

33. Complex Exponents

29

z logc z ce

When logc is specified, cz is an entire function of z.

log log log logz z c z c zd dc e e c c c

dz dz

Based on the definition, the function cz is multiple-valued. And the usual interpretation of ez (single-valued) occurs when the principal value of the logarithm is taken. The principal value of loge is unity.

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pp. 104

Ex. 2, Ex. 4, Ex. 8

33. Homework

30

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Trigonometric Functions

34. Trigonometric Functions

31

cos sin & cos sinix ixe x i x e x i x

sin & cos2 2

ix ix ix ixe e e ex x

i

Here x and y are real numbers

Based on the Euler’s Formula

sin & cos2 2

iz iz iz ize e e ez z

i

Here z is a complex number

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Trigonometric Functions

34. Trigonometric Functions

32

sin & cos2 2

iz iz iz ize e e ez z

i

Both sinz and cosz are entire since they are linear combinations of the entire Function eiz and e-iz

sin cos & cos sind d

z z z zdz dz

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pp.108-109

Ex. 2, Ex. 3

34. Homework

33

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Hyperbolic Function

35. Hyperbolic Functions

34

sinh ,cosh2 2

z z z ze e e ez z

Both sinhz and coshz are entire since they are linear combinations of the entire Function eiz and e-iz

sinh cosh , cosh sinhd d

z z z zdz dz

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Hyperbolic v.s. Trgonometric

35. Hyperbolic Functions

35

sin( ) sinh & cos( ) coshi iz z iz z

sinh( ) sin & cosh( ) cosi iz z iz z

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pp. 111-112

Ex. 3

35. Homework

36

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36. Inverse Trigonometric and Hyperbolic Functions

37

In order to define the inverse sin function sin-1z, we write1sinw z sinw zWhen

sin2

iw iwe ew z

i

2( ) 2 ( ) 1 0iw iwe iz e

2 1/2(1 )iwe iz z 1 2 1/2sin log( (1 ) )w z i iz z

Multiple-valued functions. One to infinite many values

Similar, we get 1 2 1/2cos log( (1 ) )z i z i z

1tan g2

i i zz lo

i z

Note that when specific branches of the square root and logarithmic functions are used,all three Inverse functions become single-valued and analytic.

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Inverse Hyperbolic Functions

36. Inverse Trigonometric and Hyperbolic Functions

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1 2 1/2sinh log[ ( 1) ]z z z

1 2 1/2cosh log[ ( 1) ]z z z

1 1 1tanh log

2 1

zz

z

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pp. 114-115

Ex. 1

36. Homework

39