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Chapter 3. Elementary Functions. Weiqi Luo ( 骆伟祺 ) School of Software Sun Yat-Sen University Email : [email protected] Office : # A313. Chapter 3: Elementary Functions. The Exponential Functions The Logarithmic Function Branches and Derivatives of Logarithms - PowerPoint PPT Presentation
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Chapter 3. Elementary Functions
Weiqi Luo (骆伟祺 )School of Software
Sun Yat-Sen UniversityEmail : [email protected] Office : # A313
School of Software
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
11
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
12
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
13
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
14
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
15
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
16
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
17
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
20
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
21
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
23
logc c zz e
The principal value of zc is defined by
ogc cL zz e
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33. Complex Exponents
24
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
25
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
38
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