Chapter 3. Elementary Functions
Weiqi Luo (骆伟祺)
School of Software
Sun Yat-Sen University
Email：weiqi.luo@yahoo.com Office：# A313
Chapter 3: Elementary Functions








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
iy
 cos y  i sin y
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
z1
e e
Let
e
z2
z1  z 2
z1  x1  iy1 ; z 2  x 2 +iy 2
e
x 1 +iy1
e
x 2 +iy 2
 (e 1 e
x
iy1
Real value:
x
)( e 2 e
 ( e e )(e e
x1
e
x2
x1  x 2
e
iy1
e
iy 2
x1
)
e e
x2
=e
x1  x 2
Refer to pp. 18
iy 2
)
iy
e 1e
iy 2
e
i(y 1  y 2 )
i(y 1  y 2 )
z1  z 2  ( x1  x 2 )+ i ( y1  y 2 )
z 1 +z 2
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29. The Exponential Function
 Properties
e
z1  z 2
e
z2
e
z1
e
z1
e
z2
e
z1  z 2
e
z2
Refer to Example 1 in Sec 22, (pp.68), we have that
d
e e
z
z
everywhere in the z plane
dz
which means that the function ez is entire.
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0
29. The Exponential Function
 Properties
e 0
z
e e e
z
x
iy
 re
i
For any complex number z
r  e &  y
x
r  | e | e  0 & arg( e )  y  2 n ( n  0,  1,  2, ...)
z
e
z  2 i
x
e e
z
z
2 i
e
z  2 i
 e ,e
z
2 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
e
i
 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
e e e
z
x
e 
x
x
iy

2&e
1
2e
iy
ln 2 & y 
i / 4
e

i / 4
 2 n , ( n  0,  1,  2, ...)
2
1
4
1
z  ln 2  i (  2 n ), ( n  0,  1,  2, ...)
2
4
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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 (   2 n ), ( n  0,  1,  2, ...)
z  re
i
0
Please note that the Logarithmic Function is the multiple-valued function.
ln r  i 
z  re
ln r  i (   2  )
i
One to infinite values
ln r  i (   2  )
…
It is easy to verify that
e
log z
e
ln r  i (   2 n )
e
10
ln r
e
i (   2 n )
 re
i
 z
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30. The Logarithmic Function
 The Logarithmic Function
log z  ln r  i (   2 n ), ( n  0,  1,  2, ...)
z  re
i
0
 ln | z |  i arg( z )
Suppose that 𝝝 is the principal value of argz, i.e. -π <𝝝 ≤π
L o g z  ln r  iA rg ( z )  ln r  i 
is single valued.
And
log z  L ogz  i 2 n , n  0,  1,  2, ...
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30. The Logarithmic Function
 Example 1
log(  1 
log(  1 
3 i )  log(2 e
3i )  ?
i ( 2 / 3 )
 ln 2  i ( 
)
2
 2 n ), n  0,  1,  2...
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30. The Logarithmic Function
 Example 2 & 3
log 1  ln 1  i (0  2 n )  2 n i , n  0,  1,  2, ...
L og 1  0
log(  1)  ln 1  i (  2 n )  (2 n  1) i , n  0,  1,  2, ...
L og (  1)   i
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31. Branches and Derivatives of Logarithms
 The Logarithm Function
log z  ln r  i (   2 n ), 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 ,  )  
ru r  v & u   rv r
d
log z  e
 i
dz
d
dz
L og z 
( u r  iv r )  e
 i
(
1
r
 i 0) 
1
re
i

1
z
1
z
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31. Branches and Derivatives of Logarithms
 Examples
When the principal branch is considered, then
Log ( i )  Log (  i )
3
 ln 1  i


2
And
3 L og ( i )  3(ln 1  i

)
2

i
2
3
i
2
Log ( i )  3 Log ( i )
3
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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 z 2 )  log z1  log z 2
z1  r1 e
where
i 1
 0 & z 2  r2 e
i
log( z1 z 2 )  log( r1 e 1 r2 e
i 2
i 2
0
)  ln( r1 r2 )  i ( 1   2  2 n )
 ln r1  ln r2  i ( 1  2 n1 )  i ( 2  2 n 2 )
 [ln r1  i ( 1  2 n1 )]  [ln r2   i ( 2  2 n 2 )]
 (ln | z1 |  i arg z1 )  (ln | z 2 |  i arg z 2 )
 log z1  log z 2
log(
z1
z2
n  n1  n 2
1
1
)  log( z1 z 2 )  log z1  log z 2  log z1  log z 2
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32. Some Identities Involving Logarithms
 Example
z1  z 2   1
log( z1 z 2 )  log(1)  2 n i
log( z1 )  log( z 2 )  log(  1)  (2 n  1) i
log z1  log z 2  (2 n1  1) i  (2 n 2  1) i  2( n1  n 2  1) i
 2 n i  log( z1 z 2 )
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n  n1  n 2  1
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32. Some Identities Involving Logarithms
When z≠0, then
z e
n
n log z
1
z
1/ n
e
z e
c
( n  0,  1,  2, ...)
log z
n
c 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  re i 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  exp( c log z )
c
is single-valued and analytic in the same domain.
d
dz
z 
c
d
exp( c log z ) 
dz
c
exp( c log z )
z
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33. Complex Exponents
 Example 1
i
log i  ln 1  i (
2 i

 exp(  2 i log i )
 2 n )  (2 n 
2
i
2 i
1
) i , ( n  0,  1,  2, ...)
2
 exp[(4 n  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( iL og (  i ))  exp( i (ln 1  i

2
P.V.
i
i
 exp
))  exp

2

2
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33. Complex Exponents
 Example 3
The principal branch of z2/3 can be written
exp(
2
L ogz )  exp(
3
2
ln r 
3
2
i ) 
3
3
2
r exp( i
2
)
3
Thus
2
P.V.
z3 
3
2
r cos
2
 i r sin
3
2
3
2
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 , z 2  1  i & z 3   1  i
When principal values are considered
( z1 z 2 )  2  e
i
i
z3  e
i
 / 4
e
iL og (1  i )
e
iL og (  1  i )
e
z2  e
e
iLog 2
iLog (1  i )
z1  e
i
i
 /4
e
e
3 / 4
i ln 2
( z 2 z 3 )  (  2)  e
i
iL og
（ - 2）
( z1 z 2 )  z1 z 2
i (ln 2 ) / 2
i
i (ln 2 ) / 2
e
i
i
i
( z2 z3 )  z 2 z3 e
i
i
e
i
2
i (ln 2 ) / 2
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
e
i ln 2
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
dz
c 
z
d
e
z log c
e
z log c
log c  c log c
z
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
e
ix
 cos x  i sin x & e
e e
ix
sin x 
 ix
ix
& cos x 
iz
sin z 
2i
 cos x  i sin x
e e
2i
e e
 ix
 ix
Here x and y are real numbers
2
 iz
e e
iz
& cos z 
 iz
Here z is a complex number
2
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34. Trigonometric Functions
 Trigonometric Functions
e e
iz
sin z 
 iz
e e
iz
& cos z 
2i
 iz
2
Both sinz and cosz are entire since they are linear combinations
of the entire Function eiz and e-iz
d
dz
sin z  cos z &
d
cos z   sin z
dz
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34. Homework
pp.108-109
Ex. 2, Ex. 3
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35. Hyperbolic Functions
 Hyperbolic Function
e e
z
sinh z 
z
e e
z
, cosh z 
2
z
2
Both sinhz and coshz are entire since they are linear combinations
of the entire Function eiz and e-iz
d
dz
d
sinh z  cosh z ,
cosh z  sinh z
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
1
w  sin
sin w  z 
e
iw
e
z
sin w  z
When
 iw
( e )  2 iz ( e )  1  0
iw
2i
e
iw
2
 iz  (1  z )
2 1/ 2
w  sin
Similar, we get
cos
1
tan
1
iw
1
z   i log( iz  (1  z )
2 1/ 2
z   i log( z  i (1  z )
2 1/ 2
z
i
2
lo g
i z
iz
)
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
cosh
tanh
1
z  log[ z  ( z  1)
1
z  log[ z  ( z  1)
1
2
2
z
1
2
log
1/ 2
1/ 2
]
]
1 z
1 z
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36. Homework
pp. 114-115
Ex. 1
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