Download Chap. 3 - Sun Yat

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
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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
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29. The Exponential Function
 The Exponential Function
e  e e , z  x  iy
z
x iy
Single-Valued
According to the Euler’ Formula
e  cos y  i sin y
iy
u(x,y)
v(x,y)
e  e cos y  ie sin y
z
x
x
Note that here when x=1/n (n=2,3…) & y=0, e1/n denotes the positive nth root of e.
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29. The Exponential Function
 Properties
e e e
z1
Let
z2
z1  z2
z1  x1  iy1; z2  x2 +iy2
Real value:
ex1 +iy1 ex2 +iy2  (ex1 eiy1 )(ex2 eiy2 )
 (e e )(e e )
x1 x 2
e
iy1 iy2
e x1 e x 2 =e x1  x 2
Refer to pp. 18
eiy1 eiy2  ei(y1  y2 )
x1  x 2 i(y1  y2 )
e
z1  z2  ( x1  x2 )+i( y1  y2 )
 e z1 +z2
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29. The Exponential Function
 Properties
e
z1  z2
e e
z2
e z1
z1  z2

e
z2
z2
e
e 0
z1
Refer to Example 1 in Sec 22, (pp.68), we have that
d z
e  ez
dz
everywhere in the z plane
which means that the function ez is entire.
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29. The Exponential Function
 Properties
ez  0
e  e e  re
z
x iy
i
For any complex number z
r  ex &  y
r | e z | e x  0 & arg(e z )  y  2n (n  0, 1, 2,...)
e
z  2 i
z 2 i
e e
e z 2 i  e z , e2 i  cos 2  i sin 2  1
which means that the function ez is periodic, with a pure imaginary period of 2πi
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29. The Exponential Function
 Properties
e 0
x
For any real value x
while ez can be a negative value, for instance
ei  cos   i sin   1
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29. The Exponential Function
 Example
In order to find numbers z=x+iy such that
e  1 i
z
ez  exeiy  2ei /4
i /4
e  2 &e  e
x
iy
1

ln 2 & y   2n , (n  0, 1, 2,...)
2
4
1
1
z  ln 2  i (  2n), (n  0, 1, 2,...)
2
4
x
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29. Homework
pp. 92-93
Ex. 1, Ex. 6, Ex. 8
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30. The Logarithmic Function
 The Logarithmic Function
log z  ln r  i (  2n ), (n  0, 1, 2,...)
z  rei  0
Please note that the Logarithmic Function is the multiple-valued function.
ln r  i
ln r  i(  2 )
ln r  i(  2 )
z  rei
One to infinite values
…
It is easy to verify that
elog z  eln r i (  2 n )  eln r ei (  2 n )  rei  z
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30. The Logarithmic Function
 The Logarithmic Function
log z  ln r  i (  2n ), (n  0, 1, 2,...)
z  rei  0
 ln | z | i arg( z )
Suppose that 𝝝 is the principal value of argz, i.e. -π <𝝝 ≤π
Lo g z  ln r  iArg ( z )  ln r  i
is single valued.
And
log z  Logz  i 2n , n  0, 1, 2,...
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30. The Logarithmic Function
 Example 1
log(1  3i )  ?
log(1  3i )  log(2e
i ( 2 /3)
)
2
 ln 2  i (
 2n ), n  0, 1, 2...
3
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30. The Logarithmic Function
 Example 2 & 3
log1  ln1  i(0  2n )  2n i, n  0, 1, 2,...
Log1  0
log(1)  ln1  i(  2n )  (2n  1) i, n  0, 1, 2,...
Log (1)   i
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31. Branches and Derivatives of Logarithms
 The Logarithm Function
log z  ln r  i (  2n ), n  0, 1, 2,...
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
      2
The above function becomes single-valued.
log z  ln r  i , (r  0,       2 )
With components
u (r , )  ln r & v(r , )  
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31. Branches and Derivatives of Logarithms
 The Logarithm Function
log z  ln r  i , (r  0,       2 )
is not only continuous but also analytic throughout the
domain
r  0,       2
A connected open set
  ?
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31. Branches and Derivatives of Logarithms
 The derivative of Logarithms
log z  ln r  i , (r  0,       2 )
u (r , )  ln r & v(r , )  
rur  v & u  rvr
d
1
1
 i
 i 1
log z  e (ur  ivr )  e (  i 0)  i 
dz
r
re
z
d
1
L og z 
dz
z
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31. Branches and Derivatives of Logarithms
 Examples
When the principal branch is considered, then
And
Log (i3 )  Log (i)


 ln1  i   i
2
2

3
3Log (i )  3(ln1  i ) 
i
2
2
Log (i 3 )  3Log (i)
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31. Homework
pp. 97-98
Ex. 1, Ex. 3, Ex. 4, Ex. 9, Ex. 10
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32. Some Identities Involving Logarithms
log( z1 z2 )  log z1  log z2
where
z1  r1ei1  0 & z2  r2ei2  0
log( z1 z2 )  log(r1ei1 r2ei2 )  ln(r1r2 )  i (1   2  2n )
 ln r1  ln r2  i(1  2n1 )  i(2  2n2 )
 [ln r1  i(1  2n1 )]  [ln r2  i(2  2n2 )]
 (ln | z1 | i arg z1 )  (ln | z2 | i arg z2 )
 log z1  log z2
n  n1  n2
z1
log( )  log( z1 z21 )  log z1  log z21  log z1  log z2
z2
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32. Some Identities Involving Logarithms
 Example
z1  z2  1
log( z1 z2 )  log(1)  2n i
log( z1 )  log( z2 )  log(1)  (2n  1) i
log z1  log z2  (2n1  1) i  (2n2  1) i  2(n1  n2  1) i
 2n i  log( z1 z2 )
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n  n1  n2  1
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32. Some Identities Involving Logarithms
When z≠0, then
z n  en log z (n  0, 1, 2,...)
z1/ n  e
1
log z
n
z c  ec log z
(n  1, 2,3...)
Where c is any complex number
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32. Homework
pp. 100
Ex. 1, Ex. 2, Ex. 3
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33. Complex Exponents
 Complex Exponents
When z≠0 and the exponent c is any complex number,
the function zc is defined by means of the equation
z e
c
c log z
where logz denotes the multiple-valued logarithmic
function. Thus, zc is also multiple-valued.
The principal value of zc is defined by
z e
c
cL og z
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33. Complex Exponents
If
z  rei and α is any real number, the branch
log z  ln r  i
(r  0,       2 )
Of the logarithmic function is single-valued and analytic in the indicated domain.
When the branch is used, it follows that the function
z c  exp(c log z )
is single-valued and analytic in the same domain.
d c d
c
z  exp(c log z )  exp(c log z )
dz
dz
z
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33. Complex Exponents
 Example 1
i
2i
 exp(2i log i)

1
log i  ln1  i(  2n )  (2n  ) i, ( n  0, 1, 2,...)
2
2
i
2i
 exp[(4n  1) ],(n  0, 1, 2,...)
Note that i-2i are all real numbers
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33. Complex Exponents
 Example 2
The principal value of (-i)i is


exp(iLog (i ))  exp(i (ln1  i ))  exp
2
2
P.V.
i
i  exp

2
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33. Complex Exponents
 Example 3
The principal branch of z2/3 can be written
2
2
2
2
exp( Logz )  exp( ln r  i)  3 r 2 exp(i
)
3
3
3
3
Thus
P.V.
2
3
z  3 r 2 cos
2 3 2
2
 i r sin
3
3
This function is analytic in the domain r>0, -π<𝝝<π
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33. Complex Exponents
 Example 4
Consider the nonzero complex numbers
z1  1  i, z2  1  i & z3  1  i
When principal values are considered
( z1 z2 )i  2i  eiLog 2  ei ln 2
( z2 z3 )i  (2)i  eiLog（-2） e  ei ln 2
z1i  eiLog (1i )  e /4ei (ln 2)/2
( z1 z2 )i  z1i z2i
z2i  eiLog (1i )  e /4ei (ln 2)/2
( z2 z3 )i  z2i z3i e 2
z3i  eiLog ( 1i )  e3 /4 ei (ln 2)/2
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33. Complex Exponents
 The exponential function with base c
c e
z
z log c
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.
When logc is specified, cz is an entire function of z.
d z d z log c
c  e
 e z log c log c  c z log c
dz
dz
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33. Homework
pp. 104
Ex. 2, Ex. 4, Ex. 8
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34. Trigonometric Functions
 Trigonometric Functions
Based on the Euler’s Formula
eix  cos x  i sin x & e ix  cos x  i sin x
eix  eix
eix  eix
sin x 
& cos x 
2i
2
eiz  eiz
eiz  eiz
sin z 
& cos z 
2i
2
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Here x and y are real numbers
Here z is a complex number
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34. Trigonometric Functions
 Trigonometric Functions
eiz  eiz
eiz  eiz
sin z 
& cos z 
2i
2
Both sinz and cosz are entire since they are linear combinations
of the entire Function eiz and e-iz
d
d
sin z  cos z & cos z   sin z
dz
dz
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34. Homework
pp.108-109
Ex. 2, Ex. 3
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35. Hyperbolic Functions
 Hyperbolic Function
e z  e z
e z  e z
sinh z 
, cosh z 
2
2
Both sinhz and coshz are entire since they are linear combinations
of the entire Function eiz and e-iz
d
d
sinh z  cosh z , cosh z  sinh z
dz
dz
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35. Hyperbolic Functions
 Hyperbolic v.s. Trgonometric
i sinh(iz )  sin z & cosh(iz )  cos z
i sin(iz )  sinh z & cos(iz )  cosh z
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35. Homework
pp. 111-112
Ex. 3
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36. Inverse Trigonometric and Hyperbolic Functions
In order to define the inverse sin function sin-1z, we write
w  sin 1 z
eiw  eiw
sin w  z 
2i
When
sin w  z
(eiw )2  2iz (eiw )  1  0
eiw  iz  (1  z 2 )1/2
w  sin 1 z  i log(iz  (1  z 2 )1/2 )
Similar, we get
cos1 z  i log( z  i(1  z 2 )1/2 )
i
iz
tan z  lo g
2
iz
1
Multiple-valued functions.
One to infinite many values
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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36. Inverse Trigonometric and Hyperbolic Functions
 Inverse Hyperbolic Functions
sinh 1 z  log[ z  ( z 2  1)1/2 ]
cosh 1 z  log[ z  ( z 2  1)1/2 ]
1
1 z
tanh z  log
2
1 z
1
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36. Homework
pp. 114-115
Ex. 1
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