ENGINEERING MATH. G (MATH222)
FOR ELECTRICAL AND COMPUTER ENGINEERING
Lecture 2:
Representations of Complex Numbers
(Polar and Euler’s Forms)
오성근
oskn@ajou.ac.kr
THIS LECTURE
• Polar Form of Complex Numbers
• Euler’s Form of Complex Numbers
• Some Useful Relations
• Supplementary Topics to Conjugates
• Supplementary Topics to Lecture 2
2
REPRESENTATIONS OF COMPLEX NUMBERS
(POLAR FORM)
3
POLAR FORM
• Def. The polar form of a complex number 𝑧𝑧 is defined by
𝑧𝑧 = 𝑥𝑥 + 𝑗𝑗𝑦𝑦
= 𝑟𝑟cos𝜃𝜃 + 𝑗𝑗𝑟𝑟sin𝜃𝜃
= 𝑟𝑟(cos𝜃𝜃 + 𝑗𝑗sin𝜃𝜃)
• Complex plane
Imaginary
axis
𝑗𝑗𝑗𝑗(= 𝑗𝑗𝑟𝑟 sin 𝜃𝜃)
𝑟𝑟
with 𝑟𝑟 =
𝑥𝑥 2 + 𝑦𝑦 2 and 𝜃𝜃 = tan−1
𝑥𝑥 = 𝑟𝑟 cos 𝜃𝜃 and 𝑦𝑦 = 𝑟𝑟 sin 𝜃𝜃
𝑦𝑦
𝑥𝑥
𝑧𝑧 = 𝑥𝑥 + 𝑗𝑗𝑗𝑗 = 𝑟𝑟∠𝜃𝜃
𝜃𝜃
𝑟𝑟: magnitude of a complex vector 𝑧𝑧
𝜃𝜃 : direction of the vector (counterclockwise)
Measured counterclockwise from the positive real-axis
𝑥𝑥(= 𝑟𝑟 cos θ)
Real axis
4
POLAR FORM
• Def. The polar form of a complex number 𝑧𝑧 is defined by
𝑧𝑧 = 𝑟𝑟(cos𝜃𝜃 + 𝑗𝑗sin𝜃𝜃)
In terms of the polar coordinates 𝑟𝑟, 𝜃𝜃, instead of the rectangular coordinates 𝑥𝑥, 𝑦𝑦
In short, it can be denoted as
𝑧𝑧 = 𝑟𝑟∠𝜃𝜃 with 𝑟𝑟(> 0) and 𝜃𝜃 ∈ ℝ
Imaginary
axis
𝑗𝑗𝑦𝑦
𝑟𝑟
𝑧𝑧 = 𝑥𝑥 + 𝑗𝑗𝑗𝑗
𝜃𝜃
𝑥𝑥
with 𝑟𝑟 =
Real axis
𝑥𝑥 2 + 𝑦𝑦 2 & 𝜃𝜃 = tan−1
𝑥𝑥 = 𝑟𝑟 cos 𝜃𝜃 & 𝑦𝑦 = 𝑟𝑟 sin 𝜃𝜃
𝑦𝑦
𝑥𝑥
5
POLAR FORM
• Notations
 Absolute value (or modulus) of 𝑧𝑧
𝑧𝑧 = 𝑟𝑟 =
𝑥𝑥 2 + 𝑦𝑦 2 =
𝑧𝑧 ⋅ 𝑧𝑧̅
The distance from the origin
 Argument of 𝑧𝑧
arg 𝑧𝑧
𝑦𝑦
= 𝜃𝜃 = tan−1
𝑥𝑥
Imaginary
axis
𝑗𝑗𝑦𝑦
𝑟𝑟
𝜃𝜃
Measured counterclockwise from the positive real-axis
𝑧𝑧 = 𝑥𝑥 + 𝑗𝑗𝑗𝑗
𝑥𝑥
Real axis
 𝑧𝑧 = 𝑟𝑟∠𝜃𝜃 = |z|∠arg(𝑧𝑧)
6
POLAR FORM
• Notations
• Absolute value (or modulus) of 𝑧𝑧
𝑧𝑧 = 𝑟𝑟 =
𝑥𝑥 2 + 𝑦𝑦 2 = 𝑧𝑧 ⋅ 𝑧𝑧̅
The distance from the origin
• Argument of 𝑧𝑧
arg 𝑧𝑧
𝑦𝑦
= 𝜃𝜃 = tan−1
𝑥𝑥
|𝑧𝑧|2 = 𝑧𝑧 � 𝑧𝑧̅
Imaginary
axis
𝑗𝑗𝑦𝑦
𝑟𝑟
𝜃𝜃
Measured counterclockwise from the positive real-axis
sin 𝜃𝜃 𝑟𝑟sin 𝜃𝜃 𝑦𝑦
𝑦𝑦
−1
tan 𝜃𝜃 ≡
=
= ⇒ 𝜃𝜃 = tan
cos 𝜃𝜃 𝑟𝑟cos 𝜃𝜃 𝑥𝑥
𝑥𝑥
𝑧𝑧 = 𝑥𝑥 + 𝑗𝑗𝑗𝑗
𝑥𝑥
Real axis
7
POLAR FORM
• Notations
• Absolute value (or modulus) of 𝑧𝑧
𝑧𝑧 = 𝑟𝑟 =
• Argument of 𝑧𝑧
𝑥𝑥 2 + 𝑦𝑦 2 = 𝑧𝑧𝑧𝑧̅
arg 𝑧𝑧 = 𝜃𝜃 = tan−1
• Principal value of arg(𝑧𝑧)
𝑦𝑦
𝑥𝑥
−𝜋𝜋 ≤ Arg 𝑧𝑧 < 𝜋𝜋 (※ 동영상 내용과 상이함. 앞으로 이 범위를 사용함.)
8
9
10
SIMPLE RULE FOR PHASE ADJUSTMENT (NEGATIVE REAL)
𝑦𝑦
−1
𝜃𝜃 = tan
𝑥𝑥
−1 𝑦𝑦
1) tan
𝑥𝑥
𝑦𝑦
−1
2) tan
𝑥𝑥
± 𝜋𝜋 (need adjustment if x is a negative number)
< 0, +π
> 0, −π
“𝑥𝑥”가 “음수“일 때만
11
POLAR FORM: CONJUGATE
𝑧𝑧1 = −4 − 𝑗𝑗𝑗
𝑧𝑧1 = 𝑧𝑧2 = 5
∠𝑧𝑧1
∠𝑧𝑧2
= tan−1
= tan−1
𝑧𝑧1 = 𝑟𝑟𝑟𝜃𝜃
𝑧𝑧2 = −4 + 𝑗𝑗𝑗
Note: 𝑧𝑧2 = 𝑧𝑧1̅
−3
= tan−1 0.75 − 𝜋𝜋 ≅ −2.5 𝑟𝑟𝑟𝑟𝑟𝑟
−4
3
= tan−1 −0.75 + 𝜋𝜋 ≅ 2.5 𝑟𝑟𝑟𝑟𝑟𝑟
−4
𝑧𝑧1̅ = 𝑟𝑟∠ −𝜃𝜃
공액 복소수를 극 형식(polar form)으로 나타내는 경우, 원래의 복소수와
절대값 (absolute value)은 동일하고, 인수(argument)는 부호가 반대이다.
12
POLAR FORM
• Def. Addition (and Subtraction)
 Let 𝑧𝑧1 = 𝑟𝑟1 ∠𝜃𝜃1 and z2 = 𝑟𝑟2 ∠𝜃𝜃2
 𝑧𝑧 = 𝑧𝑧1 + 𝑧𝑧2 = 𝑟𝑟1 ∠𝜃𝜃1 + 𝑟𝑟2 ∠𝜃𝜃2 (No direct computation in polar form!!!)
 Indirect computation
𝑧𝑧 = 𝑟𝑟1 ∠𝜃𝜃1 + 𝑟𝑟2 ∠𝜃𝜃2
= 𝑟𝑟1 cos𝜃𝜃1 + 𝑗𝑗sin𝜃𝜃1 + 𝑟𝑟2 (cos𝜃𝜃2 + 𝑗𝑗sin𝜃𝜃2 )
= 𝑟𝑟1 cos𝜃𝜃1 + 𝑟𝑟2 cos𝜃𝜃2 + 𝑗𝑗(𝑟𝑟1 sin𝜃𝜃1 + 𝑟𝑟2 sin𝜃𝜃2 )
=
𝑟𝑟1 cos𝜃𝜃1 + 𝑟𝑟2 cos𝜃𝜃2
2+
𝑟𝑟1 sin𝜃𝜃1 + 𝑟𝑟2 sin𝜃𝜃2
2 ∠ tan−1
𝑟𝑟1 sin𝜃𝜃1 + 𝑟𝑟2 sin𝜃𝜃2
𝑟𝑟1 cos𝜃𝜃1 + 𝑟𝑟2 cos𝜃𝜃2
polar form -> rect. form -> addition (or subtraction) in rect. form-> polar form
13
POLAR FORM
• Def. The multiplication
 Let 𝑧𝑧1 = 𝑟𝑟1 ∠𝜃𝜃1 and z2 = 𝑟𝑟2 ∠𝜃𝜃2
 𝑧𝑧1 𝑧𝑧2 = 𝑟𝑟1 ∠𝜃𝜃1 � 𝑟𝑟2 ∠𝜃𝜃2 = 𝑟𝑟1 (cos𝜃𝜃1 + 𝑗𝑗sin𝜃𝜃1 ) � 𝑟𝑟2 (cos𝜃𝜃2 + 𝑗𝑗sin𝜃𝜃2 )
= 𝑟𝑟1 𝑟𝑟2 [ cos𝜃𝜃1 cos𝜃𝜃2 − sin𝜃𝜃1 sin𝜃𝜃2 + 𝑗𝑗 cos𝜃𝜃1 sin𝜃𝜃2 + sin𝜃𝜃1 cos𝜃𝜃2 ]
= 𝑟𝑟1 𝑟𝑟2 [cos 𝜃𝜃1 + 𝜃𝜃2 + 𝑗𝑗 sin(𝜃𝜃1 + 𝜃𝜃2 )]
cos 𝛼𝛼 + 𝛽𝛽 = cos 𝛼𝛼 cos 𝛽𝛽 − sin 𝛼𝛼 sin 𝛽𝛽, sin 𝛼𝛼 + 𝛽𝛽 = sin 𝛼𝛼 cos 𝛽𝛽 + cos 𝛼𝛼 sin 𝛽𝛽
= 𝑟𝑟1 𝑟𝑟2 ∠ 𝜃𝜃1 + 𝜃𝜃2
14
POLAR FORM
• Def. The multiplication
 Let 𝑧𝑧1 = 𝑟𝑟1 ∠𝜃𝜃1 and z2 = 𝑟𝑟2 ∠𝜃𝜃2
 𝑧𝑧1 𝑧𝑧2 = 𝑟𝑟1 𝑟𝑟2 cos 𝜃𝜃1 + 𝜃𝜃2 + 𝑗𝑗 sin 𝜃𝜃1 + 𝜃𝜃2
= 𝑟𝑟1 𝑟𝑟2 ∠ 𝜃𝜃1 + 𝜃𝜃2
• Properties
 Absolute value: 𝑧𝑧1 𝑧𝑧2 = 𝑧𝑧1 𝑧𝑧2
 Argument value: arg 𝑧𝑧1 𝑧𝑧2 = arg 𝑧𝑧1 + arg(𝑧𝑧2 )
15
POLAR FORM
• Def. The division
 Let 𝑧𝑧1 = 𝑟𝑟1 ∠𝜃𝜃1 and z2 = 𝑟𝑟2 ∠𝜃𝜃2
𝑧𝑧1
= 𝑟𝑟1 ∠𝜃𝜃1
𝑧𝑧2
= 𝑟𝑟1 ∠𝜃𝜃1
=
1
𝑟𝑟2 ∠𝜃𝜃2
1
∠(−𝜃𝜃2 )
𝑟𝑟2
𝑟𝑟1
[cos 𝜃𝜃1 − 𝜃𝜃2 + 𝑗𝑗 sin(𝜃𝜃1 − 𝜃𝜃2 )]
𝑟𝑟2
16
1
1
=
𝑧𝑧2 𝑟𝑟2 (cos 𝜃𝜃2 + 𝑗𝑗 sin 𝜃𝜃2 )
(cos 𝜃𝜃2 − 𝑗𝑗 sin 𝜃𝜃2 )
=
𝑟𝑟2 (cos 𝜃𝜃2 + 𝑗𝑗 sin 𝜃𝜃2 )(cos 𝜃𝜃2 − 𝑗𝑗 sin 𝜃𝜃2 )
cos 𝜃𝜃2 − 𝑗𝑗 sin 𝜃𝜃2
=
𝑟𝑟2 [(cos 𝜃𝜃2 )2 + (sin 𝜃𝜃2 )2 ]
cos(−𝜃𝜃2 ) + 𝑗𝑗 sin(− 𝜃𝜃2 )
𝑟𝑟2 [(cos 𝜃𝜃2 )2 + (sin 𝜃𝜃2 )2 ]
1
= ∠(−𝜃𝜃2 )
𝑟𝑟2
=
or
1
𝑧𝑧2̅
𝑟𝑟2 ∠(−𝜃𝜃2 ) 1
=
=
= ∠(−𝜃𝜃2 )
2
𝑧𝑧2 𝑧𝑧2 𝑧𝑧2̅
𝑟𝑟2
𝑟𝑟2
17
POLAR FORM
• Def. The division
 Let 𝑧𝑧1 = 𝑟𝑟1 ∠𝜃𝜃1 and z2 = 𝑟𝑟2 ∠𝜃𝜃2

𝑧𝑧1
𝑟𝑟
= 1 [cos 𝜃𝜃1 − 𝜃𝜃2
𝑧𝑧2
𝑟𝑟2
• Properties
 Absolute value:
𝑧𝑧1
𝑧𝑧2
=
 Argument value: arg
+ 𝑗𝑗 sin(𝜃𝜃1 − 𝜃𝜃2 )]
𝑧𝑧1
𝑧𝑧2
𝑧𝑧1
𝑧𝑧2
= arg 𝑧𝑧1 − arg(𝑧𝑧2 )
18
REPRESENTATIONS OF COMPLEX NUMBERS
(EULER’S FORM)
19
EULER’S FORM
• [Fact] Euler’s formula
𝑒𝑒 𝑗𝑗𝜃𝜃 = cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃
where 𝑒𝑒 is the base of the natural logarithm
(The proof is covered in page 46.)
The fundamental relationship between the trigonometric
functions and the complex exponential function
20
EULER’S EQUATIONS
• Basic Formulas
𝑒𝑒 𝑗𝑗𝑗𝑗 = cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃
𝑒𝑒 −𝑗𝑗𝑗𝑗 = cos(−𝜃𝜃) + 𝑗𝑗sin −𝜃𝜃
• Derivations
𝑒𝑒 𝑗𝑗𝑗𝑗 + 𝑒𝑒 −𝑗𝑗𝑗𝑗
cos(𝜃𝜃) =
2
𝑒𝑒 𝑗𝑗𝑗𝑗 − 𝑒𝑒 −𝑗𝑗𝑗𝑗
sin(𝜃𝜃) =
2𝑗𝑗
= cos(𝜃𝜃) − 𝑗𝑗sin 𝜃𝜃
2𝑗𝑗sin(𝜃𝜃)
= 𝑒𝑒 𝑗𝑗𝑗𝑗 =
1
𝑒𝑒 𝑗𝑗𝑗𝑗
2cos(𝜃𝜃)
21
EULER’S FORM
• Def. The Euler’s form of a complex number 𝑧𝑧 is defined as
𝑧𝑧 = 𝑟𝑟𝑒𝑒 𝑗𝑗𝜃𝜃 with 𝑟𝑟(> 0) and 𝜃𝜃 ∈ ℝ
(⇐ 𝑧𝑧 = 𝑟𝑟∠𝜃𝜃 in polar form)
• Relations
𝑟𝑟𝑒𝑒 𝑗𝑗𝜃𝜃 = 𝑟𝑟 cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃 = 𝑥𝑥 + 𝑗𝑗𝑗𝑗 =
(Warning)
𝜃𝜃 = tan−1
𝑦𝑦
𝑥𝑥
𝑥𝑥 2 + 𝑦𝑦 2 𝑒𝑒
𝑟𝑟
𝑦𝑦
𝑗𝑗tan−1 𝑥𝑥
𝜃𝜃
± 𝜋𝜋 (need adjustment if x is a negative number)
22
REPRESENTATIONS: SUMMARY
• A complex number 𝑧𝑧 ∈ ℂ is represented by
 Rectangular form
𝑧𝑧 = 𝑥𝑥 + 𝑗𝑗𝑗𝑗 (𝑥𝑥, 𝑦𝑦 ∈ ℝ)
 Polar form
𝑧𝑧 = 𝑟𝑟 cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃 (𝑟𝑟(> 0), 𝜃𝜃 ∈ ℝ)
= 𝑟𝑟∠𝜃𝜃 (𝑟𝑟(> 0), 𝜃𝜃 ∈ ℝ)
 Euler’s form
𝑧𝑧 = 𝑟𝑟𝑒𝑒 𝑗𝑗𝜃𝜃 (𝑟𝑟(> 0), 𝜃𝜃 ∈ ℝ)
23
REPRESENTATIONS: CONVERSION DIAGRAM
Euler’s form
𝜃𝜃 = tan−1
Rectangular form
𝑦𝑦
𝑥𝑥
± 𝜋𝜋 (need adjustment if x is a negative number)
24
EULER’S FORM
• Multiplication and Division
 Let 𝑧𝑧1 = 𝑥𝑥1 + 𝑗𝑗𝑦𝑦1 = 𝑟𝑟1 𝑒𝑒 𝑗𝑗𝜃𝜃1 and 𝑧𝑧2 = 𝑥𝑥2 + 𝑗𝑗𝑦𝑦2 = 𝑟𝑟2 𝑒𝑒 𝑗𝑗𝜃𝜃2
𝑧𝑧1 ⋅ 𝑧𝑧2 = 𝑟𝑟1 𝑒𝑒 𝑗𝑗𝜃𝜃1 𝑟𝑟2 𝑒𝑒 𝑗𝑗𝜃𝜃2 = 𝑟𝑟1 𝑟𝑟2 𝑒𝑒 𝑗𝑗(𝜃𝜃1+𝜃𝜃2)
𝑧𝑧1 𝑟𝑟1 𝑒𝑒 𝑗𝑗𝜃𝜃1 𝑟𝑟1 𝑒𝑒 𝑗𝑗𝜃𝜃1 𝑒𝑒 −𝑗𝑗𝜃𝜃2 𝑟𝑟1 𝑗𝑗(𝜃𝜃 −𝜃𝜃 )
=
=
= 𝑒𝑒 1 2
𝑗𝑗𝜃𝜃
𝑧𝑧2 𝑟𝑟2 𝑒𝑒 2
𝑟𝑟2
𝑟𝑟2
 Let 𝑧𝑧 = 𝑥𝑥 + 𝑗𝑗𝑗𝑗 = 𝑟𝑟𝑒𝑒 𝑗𝑗𝜃𝜃
𝑧𝑧 𝑛𝑛 = 𝑟𝑟𝑒𝑒 𝑗𝑗𝜃𝜃
𝑧𝑧1/𝑛𝑛 =
𝑛𝑛
= 𝑟𝑟 𝑛𝑛 𝑒𝑒 𝑗𝑗𝑛𝑛𝜃𝜃
1
1/𝑛𝑛
𝑗𝑗𝜃𝜃
𝑟𝑟𝑒𝑒
= 𝑟𝑟 𝑛𝑛
cos
𝜃𝜃+2𝜋𝜋𝑘𝑘
𝑛𝑛
+ 𝑗𝑗 sin
𝑘𝑘 = 0,1,2, … , (𝑛𝑛 − 1)
𝜃𝜃+2𝜋𝜋𝑘𝑘
𝑛𝑛
(later)
25
REPRESENTATIONS AND OPERATIONS
• Let 𝑧𝑧1 = 𝑥𝑥1 + 𝑗𝑗𝑦𝑦1 = 𝑟𝑟1 𝑒𝑒 𝑗𝑗𝜃𝜃1 and 𝑧𝑧2 = 𝑥𝑥2 + 𝑗𝑗𝑦𝑦2 = 𝑟𝑟2 𝑒𝑒 𝑗𝑗𝜃𝜃2
Addition and Subtraction
𝑧𝑧1 ± 𝑧𝑧2 = 𝑥𝑥1 ± 𝑥𝑥2 + 𝑗𝑗(𝑦𝑦1 ± 𝑦𝑦2 )
Multiplication and Division
𝑧𝑧1 ⋅ 𝑧𝑧2 = 𝑟𝑟1 𝑒𝑒
𝑗𝑗𝜃𝜃1
𝑟𝑟2 𝑒𝑒
𝑗𝑗𝜃𝜃2
= 𝑟𝑟1 𝑟𝑟2 𝑒𝑒
𝑗𝑗(𝜃𝜃1 +𝜃𝜃2 )
𝑧𝑧1 𝑟𝑟1 𝑒𝑒 𝑗𝑗𝜃𝜃1 𝑟𝑟1 𝑗𝑗(𝜃𝜃 −𝜃𝜃 )
=
= 𝑒𝑒 1 2
𝑗𝑗𝜃𝜃
𝑧𝑧2 𝑟𝑟2 𝑒𝑒 2 𝑟𝑟2
Conjugation
𝑧𝑧1∗ = 𝑧𝑧1̅ = 𝑥𝑥1 − 𝑗𝑗𝑦𝑦1 = 𝑟𝑟1 𝑒𝑒 −𝑗𝑗𝜃𝜃1
𝑧𝑧1 − 𝑧𝑧2 =
Distance
𝑥𝑥1 − 𝑥𝑥2 2 + 𝑦𝑦1 − 𝑦𝑦2 2
Unit Circle
𝑧𝑧: 𝑧𝑧 = 𝑒𝑒 𝑗𝑗𝑗𝑗 , −𝜋𝜋 ≤ 𝜃𝜃 < 𝜋𝜋
26
REPRESENTATIONS AND OPERATIONS
• Unit Circle
 𝑒𝑒 𝑗𝑗𝜃𝜃 = cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃
−𝜋𝜋 ≤ 𝜃𝜃 < 𝜋𝜋
|𝑒𝑒 𝑗𝑗𝑗𝑗 | =
(cos 𝜃𝜃)2 + sin 𝜃𝜃 2 = 1
27
MULTIPLICATION OF COMPLEX NUMBERS
 Let 𝒛𝒛𝟏𝟏 = 𝒙𝒙𝟏𝟏 + 𝒋𝒋𝒚𝒚𝟏𝟏 and 𝒛𝒛𝟐𝟐 = 𝒙𝒙𝟐𝟐 + 𝒋𝒋𝒚𝒚𝟐𝟐
 Rectangular Form
𝑧𝑧1 ⋅ 𝑧𝑧2 = 𝑥𝑥1 + 𝑗𝑗𝑦𝑦1 𝑥𝑥2 + 𝑗𝑗𝑦𝑦2
= 𝑥𝑥1 𝑥𝑥2 + 𝑗𝑗𝑥𝑥1 𝑦𝑦2 + 𝑗𝑗𝑦𝑦1 𝑥𝑥2 + 𝑗𝑗 2 𝑦𝑦1 𝑦𝑦2
= 𝑥𝑥1 𝑥𝑥2 − 𝑦𝑦1 𝑦𝑦2 + 𝑗𝑗(𝑥𝑥1 𝑦𝑦2 + 𝑦𝑦1 𝑥𝑥2 )
 Euler’s Form
𝑧𝑧1 ⋅ 𝑧𝑧2 = 𝑟𝑟1 𝑒𝑒 𝑗𝑗𝜃𝜃1 � 𝑟𝑟2 𝑒𝑒 𝑗𝑗𝜃𝜃2
=
=
𝑗𝑗 tan
𝑥𝑥12 + 𝑦𝑦12 𝑒𝑒
𝑥𝑥12 + 𝑦𝑦12
−1 𝑦𝑦1
𝑥𝑥1
�
𝑗𝑗 tan
𝑥𝑥22 + 𝑦𝑦22 𝑒𝑒
−1 𝑦𝑦2
𝑥𝑥2
−1 𝑦𝑦1 +tan−1 𝑦𝑦2
𝑗𝑗
tan
𝑥𝑥1
𝑥𝑥2
𝑥𝑥22 + 𝑦𝑦22 𝑒𝑒
28
DIVISION OF COMPLEX NUMBERS
 Let 𝒛𝒛𝟏𝟏 = 𝒙𝒙𝟏𝟏 + 𝒋𝒋𝒚𝒚𝟏𝟏 and 𝒛𝒛𝟐𝟐 = 𝒙𝒙𝟐𝟐 + 𝒋𝒋𝒚𝒚𝟐𝟐
 Rectangular Form
𝑧𝑧 =
𝑧𝑧1 (𝑥𝑥1 + 𝑗𝑗𝑦𝑦1 )(𝑥𝑥2 − 𝑗𝑗𝑦𝑦2 ) 𝑥𝑥1 𝑥𝑥2 + 𝑦𝑦1 𝑦𝑦2
−𝑥𝑥1 𝑦𝑦2 + 𝑦𝑦1 𝑥𝑥2
=
=
+
𝑗𝑗
𝑧𝑧2 (𝑥𝑥2 + 𝑗𝑗𝑦𝑦2 )(𝑥𝑥2 − 𝑗𝑗𝑦𝑦2 )
𝑥𝑥22 + 𝑦𝑦22
𝑥𝑥22 + 𝑦𝑦22
 Euler’s Form
𝑧𝑧1 𝑟𝑟1 𝑒𝑒 𝑗𝑗𝜃𝜃1 𝑟𝑟1 𝑗𝑗(𝜃𝜃 −𝜃𝜃 )
𝑧𝑧 = =
= 𝑒𝑒 1 2
𝑗𝑗𝜃𝜃
𝑧𝑧2 𝑟𝑟2 𝑒𝑒 2 𝑟𝑟2
=
−1 𝑦𝑦1
𝑗𝑗
tan
𝑥𝑥1
𝑥𝑥12 + 𝑦𝑦12 𝑒𝑒
−1 𝑦𝑦2
2
2 𝑗𝑗 tan
𝑥𝑥2
𝑥𝑥2 + 𝑦𝑦2 𝑒𝑒
=
𝑥𝑥12 + 𝑦𝑦12
𝑥𝑥22 + 𝑦𝑦22
𝑒𝑒
𝑦𝑦
𝑦𝑦
𝑗𝑗 tan−1 𝑥𝑥1 −tan−1 𝑥𝑥2
1
2
29
SOME USEFUL RELATIONS
30
TRIANGLE INEQUALITY
• Triangle Inequality
𝑧𝑧1 + 𝑧𝑧2 ≤ 𝑧𝑧1 + 𝑧𝑧2
Imaginary
axis
𝑧𝑧1
𝑧𝑧2
𝑧𝑧1 + 𝑧𝑧2
Real axis
31
GENERALIZED TRIANGLE INEQUALITY
• Triangle Inequality
𝑧𝑧1 + 𝑧𝑧2 ≤ 𝑧𝑧1 + 𝑧𝑧2
• Generalized Triangle Inequality
𝑛𝑛
𝑛𝑛
𝑘𝑘=1
𝑘𝑘=1
� 𝑧𝑧𝑘𝑘 ≤ � 𝑧𝑧𝑘𝑘
32
DE MOIVRE’S FORMULA
• De Moivre’s Formula
𝑧𝑧 𝑛𝑛 =
𝑛𝑛
𝑗𝑗𝑗𝑗
𝑟𝑟𝑒𝑒
= 𝑟𝑟 𝑛𝑛 𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗 = 𝑟𝑟 𝑛𝑛 (cos 𝑛𝑛𝑛𝑛 + 𝑗𝑗 sin 𝑛𝑛𝑛𝑛)
(𝑒𝑒 𝑗𝑗𝑗𝑗 )𝑛𝑛 = cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃 𝑛𝑛 = cos 𝑛𝑛𝑛𝑛 + 𝑗𝑗 sin 𝑛𝑛𝑛𝑛 = 𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗
• [Example] Compute the following using the above formula
cos 2𝜃𝜃 =?
sin 2𝜃𝜃 =?
[Hint: cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃 𝑛𝑛 = cos 𝑛𝑛𝑛𝑛 + 𝑗𝑗 sin 𝑛𝑛𝑛𝑛 ]
33
DE MOIVRE’S FORMULA
• De Moivre’s Formula
𝑧𝑧 𝑛𝑛 = 𝑟𝑟𝑒𝑒 𝑗𝑗𝑗𝑗
𝑛𝑛
= 𝑟𝑟 𝑛𝑛 𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗 = 𝑟𝑟 𝑛𝑛 (cos 𝑛𝑛𝑛𝑛 + 𝑗𝑗 sin 𝑛𝑛𝑛𝑛)
(𝑒𝑒 𝑗𝑗𝑗𝑗 )𝑛𝑛 = cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃 𝑛𝑛 = cos 𝑛𝑛𝑛𝑛 + 𝑗𝑗 sin 𝑛𝑛𝑛𝑛 = 𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗
• [Example] Compute the following using the above formula
cos 2𝜃𝜃 =?,
sin 2𝜃𝜃 =?
cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃 2 = cos 2𝜃𝜃 + 𝑗𝑗 sin 2𝜃𝜃
cos 2 𝜃𝜃 − sin2 𝜃𝜃 + 𝑗𝑗 2cos 𝜃𝜃 sin 𝜃𝜃 = cos 2𝜃𝜃 + 𝑗𝑗sin2𝜃𝜃
cos 2𝜃𝜃 = cos 2 𝜃𝜃 − sin2 𝜃𝜃
sin2𝜃𝜃 = 2cos 𝜃𝜃 sin 𝜃𝜃
34
SUPPLEMENTARY TOPICS TO CONJUGATES
35
CONJUGATES
𝑗𝑗𝜃𝜃
𝑧𝑧 = 𝑥𝑥 + 𝑗𝑗𝑗𝑗 = 𝑟𝑟𝑒𝑒 , with 𝑟𝑟 =
𝑟𝑟 =
𝑟𝑟 =
𝑥𝑥 2 + 𝑦𝑦 2 and 𝜃𝜃 = tan−1
𝑧𝑧 = 𝑥𝑥 + 𝑗𝑗𝑦𝑦 = 𝑟𝑟𝑒𝑒 𝑗𝑗𝜃𝜃 ,
𝑧𝑧̅ = 𝑥𝑥 − 𝑗𝑗𝑗𝑗 = 𝑟𝑟𝑒𝑒 −𝑗𝑗𝜃𝜃
𝑥𝑥 2 + 𝑦𝑦 2 and 𝜃𝜃 = tan−1
𝑦𝑦
𝑥𝑥
𝑦𝑦
−1
2
2
𝑥𝑥 + 𝑦𝑦 and 𝜃𝜃 = tan
𝑥𝑥
𝑦𝑦
𝑥𝑥
36
CONJUGATES
𝑧𝑧 = 𝑥𝑥 + 𝑗𝑗𝑗𝑗 = 𝑟𝑟𝑒𝑒 𝑗𝑗𝜃𝜃
- 𝑧𝑧 + 𝑧𝑧̅ = 2𝑥𝑥 = 2Re 𝑧𝑧
- 𝑧𝑧 − 𝑧𝑧̅ = 𝑗𝑗𝑗𝑗𝑗 = 𝑗𝑗2Im 𝑧𝑧
𝑟𝑟 =
𝑥𝑥 2 + 𝑦𝑦 2 and 𝜃𝜃 = tan−1
𝑥𝑥 = 𝑟𝑟 cos 𝜃𝜃 and 𝑦𝑦 = 𝑟𝑟 sin 𝜃𝜃
𝑦𝑦
𝑥𝑥
- 𝑧𝑧 ⋅ 𝑧𝑧̅ = 𝑥𝑥 2 + 𝑦𝑦 2
- 𝑧𝑧 + 𝑧𝑧̅ = 𝑟𝑟 𝑒𝑒 𝑗𝑗𝜃𝜃 + 𝑒𝑒 −𝑗𝑗𝜃𝜃 = 2𝑟𝑟 cos 𝜃𝜃 = 2𝑟𝑟Re 𝑒𝑒 𝑗𝑗𝜃𝜃
- 𝑧𝑧 − 𝑧𝑧̅ = 𝑟𝑟 𝑒𝑒 𝑗𝑗𝜃𝜃 − 𝑒𝑒 −𝑗𝑗𝜃𝜃 = 𝑗𝑗𝑗𝑗𝑗 sin 𝜃𝜃 = 𝑗𝑗𝑗𝑗𝑗Im 𝑒𝑒 𝑗𝑗𝜃𝜃
- 𝑧𝑧 ⋅ 𝑧𝑧̅ = 𝑟𝑟𝑒𝑒 𝑗𝑗𝜃𝜃 � 𝑟𝑟𝑒𝑒 −𝑗𝑗𝜃𝜃 = 𝑟𝑟 2
37
CONJUGATES
𝑧𝑧1 = 𝑥𝑥1 + 𝑗𝑗𝑦𝑦1 = 𝑟𝑟1 𝑒𝑒 𝑗𝑗𝜃𝜃1 , 𝑧𝑧2 = 𝑥𝑥2 + 𝑗𝑗𝑦𝑦2 = 𝑟𝑟2 𝑒𝑒 𝑗𝑗𝜃𝜃2 , 𝑧𝑧 = 𝑥𝑥 + 𝑗𝑗𝑗𝑗 = 𝑟𝑟𝑒𝑒 𝑗𝑗𝜃𝜃
- 𝑧𝑧1 + 𝑧𝑧2 = 𝑧𝑧�1 + 𝑧𝑧�2
- 𝑧𝑧1 𝑧𝑧2 = 𝑧𝑧�1 𝑧𝑧�2
-
𝑍𝑍1
𝑍𝑍2
=
- 𝑧𝑧̿ = 𝑧𝑧
𝑍𝑍1
𝑍𝑍2
38
CONJUGATES
𝑧𝑧 = 𝑥𝑥 + 𝑗𝑗𝑗𝑗 = 𝑟𝑟𝑒𝑒 𝑗𝑗𝜃𝜃
- 𝑠𝑠 − 𝑧𝑧 𝑠𝑠 − 𝑧𝑧̅ = 𝑠𝑠 2 − 𝑠𝑠𝑧𝑧̅ − 𝑠𝑠𝑠𝑠 + 𝑧𝑧𝑧𝑧̅ = 𝑠𝑠 2 − 2Re[𝑧𝑧]𝑠𝑠 + |𝑧𝑧|2
= [𝑠𝑠 2 − 2𝑥𝑥𝑥𝑥 + 𝑥𝑥 2 + 𝑦𝑦 2 ]
⟹ all coefficients are real
= [𝑠𝑠 2 − 2𝑟𝑟 cos 𝜃𝜃 𝑠𝑠 + 𝑟𝑟 2 ]
실계수 2차 방정식( 𝑎𝑎𝑥𝑥 2 +𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0 (𝑎𝑎, 𝑏𝑏, 𝑐𝑐: real))이 복소근을 가질 경우,
두 근은 상호 복소수 공액 쌍(complex conjugate pair)을 이룬다.
𝑏𝑏
𝑥𝑥1,2 = −
± 𝑗𝑗
2𝑎𝑎
𝑏𝑏 2 − 4𝑎𝑎𝑎𝑎
2𝑎𝑎
39
SUPPLEMENTARY TOPICS TO LECTURE 2
40
⋯
-2𝜋𝜋
⋯
0
0 ≤ 𝜃𝜃 < 2𝜋𝜋
0
+𝜋𝜋
0
-𝜋𝜋
−𝜋𝜋 ≤ 𝜃𝜃 < 𝜋𝜋
arg 𝑧𝑧 = 𝜃𝜃 = Arg 𝑧𝑧 + 2𝑛𝑛𝜋𝜋 (𝑛𝑛: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)
𝜋𝜋
2
⋯
arz 𝑧𝑧 = 𝜃𝜃
𝜋𝜋
2𝜋𝜋
2𝜋𝜋
4𝜋𝜋
𝜃𝜃
⋯
0 2𝜋𝜋
3𝜋𝜋
2
https://math.libretexts.org/Bookshelves/Linear_Algebra/Complex_Analysis_-_A_Visual_and_Interactive_Introduction_(Ponce_Campuzano)/01%3A_Chapter_1/1.04%3A_The_Principal_Argument
SUPPLEMENTARY TOPICS
• Sum of Inverse Tangents
If 𝑥𝑥1 , 𝑥𝑥2 , … , 𝑥𝑥𝑛𝑛 ∈ 𝑅𝑅, then
tan−1 𝑥𝑥1 + tan−1 𝑥𝑥2 + ⋯ + tan−1 𝑥𝑥𝑛𝑛 = tan−1
𝑠𝑠1 − 𝑠𝑠3 + 𝑠𝑠5 − 𝑠𝑠7 + ⋯
1 − 𝑠𝑠2 + 𝑠𝑠4 − 𝑠𝑠6 + ⋯
where 𝑠𝑠𝑘𝑘 = 𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜𝑜𝑜 𝑥𝑥𝑥, 𝑥𝑥𝑥, … 𝑥𝑥𝑛𝑛 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑘𝑘 𝑎𝑎𝑎𝑎 𝑎𝑎 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡.
For example, tan−1 𝑥𝑥 + tan−1 𝑦𝑦 + tan−1 𝑧𝑧 = tan−1
Note: −tan−1 𝑥𝑥 = tan−1 (−𝑥𝑥).
𝑥𝑥+𝑦𝑦+𝑧𝑧 −𝑥𝑥𝑥𝑥𝑥𝑥
1−(𝑥𝑥𝑥𝑥+𝑥𝑥𝑥𝑥+𝑦𝑦𝑦𝑦)
• Sum of Two Inverse Tangents
tan−1 𝑥𝑥
+ tan−1 𝑦𝑦
= tan−1
𝑥𝑥 + 𝑦𝑦
1 − 𝑥𝑥𝑥𝑥
42
SUPPLEMENTARY TOPICS
• Sum of Two Inverse Tangents
−1
−1
tan (𝑥𝑥) + tan
Proof)
−1
𝑦𝑦 = tan
𝑥𝑥 + 𝑦𝑦
1 − 𝑥𝑥𝑥𝑥
tan−1 (𝑥𝑥) = 𝐴𝐴, tan−1 𝑦𝑦 = 𝐵𝐵
⇒ tan 𝐴𝐴 = 𝑥𝑥, tan 𝐵𝐵 = 𝑦𝑦
From trigonometric identity, tan 𝐴𝐴 + 𝐵𝐵 =
⇒ 𝐴𝐴 + 𝐵𝐵 = tan−1
tan 𝐴𝐴+tan 𝐵𝐵
1−tan 𝐴𝐴 tan 𝐵𝐵
⇒ tan−1 (𝑥𝑥) + tan−1 (𝑦𝑦) = tan−1
tan 𝐴𝐴+tan 𝐵𝐵
1−tan 𝐴𝐴 tan 𝐵𝐵
𝑥𝑥+𝑦𝑦
1−𝑥𝑥𝑥𝑥
43
DE MOIVRE’S THEOREM
• De Moivre’s Theorem
𝑧𝑧 𝑛𝑛 = 𝑟𝑟𝑒𝑒 𝑗𝑗𝑗𝑗
Proof
𝑛𝑛
= 𝑟𝑟 𝑛𝑛 𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗 = 𝑟𝑟 𝑛𝑛 (cos 𝑛𝑛𝑛𝑛 + 𝑗𝑗 sin 𝑛𝑛𝑛𝑛)
(𝑒𝑒 𝑗𝑗𝑗𝑗 )𝑛𝑛 = cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃 𝑛𝑛 = cos 𝑛𝑛𝑛𝑛 + 𝑗𝑗 sin 𝑛𝑛𝑛𝑛 = 𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗
𝑛𝑛 = 1
cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃 1 = cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃
𝑛𝑛 = 𝑘𝑘
cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃 𝑘𝑘 = cos 𝑘𝑘𝜃𝜃 + 𝑗𝑗 sin 𝑘𝑘𝜃𝜃
𝑛𝑛 = 𝑘𝑘 + 1
cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃 𝑘𝑘+1 = cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃 𝑘𝑘 (cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃)
= cos 𝑘𝑘𝜃𝜃 + 𝑗𝑗 sin 𝑘𝑘𝜃𝜃 (cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃)
= cos 𝑘𝑘𝜃𝜃 cos 𝜃𝜃 − sin 𝑘𝑘𝜃𝜃 sin 𝜃𝜃 + 𝑗𝑗(sin 𝑘𝑘𝜃𝜃 cos 𝜃𝜃 + cos 𝑘𝑘𝜃𝜃 sin 𝜃𝜃)
= cos(𝑘𝑘𝜃𝜃 + 𝜃𝜃) + 𝑗𝑗 sin(𝑘𝑘𝜃𝜃 + 𝜃𝜃) = cos 𝑘𝑘 + 1 𝜃𝜃 + 𝑗𝑗 sin 𝑘𝑘 + 1 𝜃𝜃
Note: cos(𝛼𝛼 + 𝛽𝛽) = cos 𝛼𝛼 cos 𝛽𝛽 − sin 𝛼𝛼 sin 𝛽𝛽, sin(𝛼𝛼 + 𝛽𝛽) = sin 𝛼𝛼 cos 𝛽𝛽 + cos 𝛼𝛼 sin 𝛽𝛽
44
DE MOIVRE’S AND N-TH ROOT THEOREMS
• De Moivre’s Theorem
𝑧𝑧 𝑛𝑛 =
•
𝑛𝑛
𝑗𝑗𝑗𝑗
𝑟𝑟𝑒𝑒
= 𝑟𝑟 𝑛𝑛 𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗 = 𝑟𝑟 𝑛𝑛 (cos 𝑛𝑛𝑛𝑛 + 𝑗𝑗 sin 𝑛𝑛𝑛𝑛)
(𝑒𝑒 𝑗𝑗𝑗𝑗 )𝑛𝑛 = cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃 𝑛𝑛 = cos 𝑛𝑛𝑛𝑛 + 𝑗𝑗 sin 𝑛𝑛𝑛𝑛 = 𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗
De Moivre’s n-th Root Theorem
𝑧𝑧1/𝑛𝑛 =
1
1/𝑛𝑛
𝑗𝑗𝑗𝑗
𝑟𝑟𝑒𝑒
= 𝑟𝑟 𝑛𝑛
𝜃𝜃 + 2𝜋𝜋𝑘𝑘
𝜃𝜃 + 2𝜋𝜋𝑘𝑘
cos
+ 𝑗𝑗 sin
𝑛𝑛
𝑛𝑛
𝑘𝑘 = 0,1,2, … , (𝑛𝑛 − 1)
45
EULER’S THEOREM
2
3
4
5
6
7
𝑥𝑥
𝑥𝑥
𝑥𝑥
𝑥𝑥
𝑥𝑥
𝑥𝑥
+ + + ⋯
𝑒𝑒 𝑥𝑥 = 1 + 𝑥𝑥 + + +
2! 3! 4! 5! 6! 7!
𝑥𝑥 2 𝑥𝑥 4 𝑥𝑥 6
cos 𝑥𝑥 = 1 − + − ⋯
2! 4! 6!
𝑥𝑥 3 𝑥𝑥 5 𝑥𝑥 7
+ − ⋯
sin 𝑥𝑥 = 𝑥𝑥 −
3! 5! 7!
𝑥𝑥 → 𝑗𝑗𝜃𝜃
2 𝜃𝜃 2
3 𝜃𝜃 3
4 𝜃𝜃 4
5 𝜃𝜃 5
6 𝜃𝜃 6
7 𝜃𝜃 7
𝑗𝑗
𝑗𝑗
𝑗𝑗
𝑗𝑗
𝑗𝑗
𝑗𝑗
+
+
+
+
+
⋯
𝑒𝑒 𝑗𝑗𝜃𝜃 = 1 + 𝑗𝑗𝜃𝜃 +
2!
3!
4!
5!
6!
7!
𝜃𝜃 2
𝜃𝜃 3 𝜃𝜃 4
𝜃𝜃 5 𝜃𝜃 6
𝜃𝜃 7
= 1 + 𝑗𝑗𝜃𝜃 −
− 𝑗𝑗 +
+ 𝑗𝑗 −
− 𝑗𝑗 ⋯
2!
3! 4!
5! 6!
7!
𝜃𝜃 2 𝜃𝜃 4 𝜃𝜃 6
𝜃𝜃 3 𝜃𝜃 5 𝜃𝜃 7
+
− ⋯ + 𝑗𝑗 𝜃𝜃 −
+
− ⋯
= 1−
2! 4! 6!
3! 5! 7!
= cos 𝜃𝜃 + 𝑗𝑗 sin 𝜃𝜃