Complex
Variables &
Transforms
MATH- 232
Instructor: Dr. Naila Amir
Complex Numbers
Complex Numbers
Book: A First Course in Complex Analysis with Applications by
Dennis G. Zill and Patrick D. Shanahan.
• Chapter: 1
• Sections: 1.1, 1.2
Complex numbers
▪ Complex Numbers and Their Properties
▪ Complex Plane
▪ Polar Form of Complex Numbers
▪ Powers and Roots
▪ Sets of Points in the Complex Plane
▪ Applications
Complex Numbers
• Complex Numbers show up all over the place in engineering, computer science
and as well as scientific computing. Examples include:
• Fast Fourier Transforms for Signal Processing,
• Circuit Simulation (Complex Numbers are very common in Electrical
Engineering),
• Fractals which get used in Graphics and various other fields.
• Closely related to the electrical engineering, example is the use of complex
numbers in signal processing, which has applications to:
• telecommunications (cellular phone),
• radar (which assists the navigation of airplanes).
How is Complex Analysis Useful in Engineering?
Almost everything in engineering involves complex analysis at some level:
▪ Basic circuit analysis using Laplace transforms and phasors.
▪ The concept of complex impedance.
▪ Conformal maps like the Smith Chart.
▪ Digital modulation and analysis of its performance etc.
▪ Complex analysis is used in any kind of circuit design including power systems
and electronics and is also completely fundamental to signal processing and
communications.
Complex Numbers
In order to define a complex number, we first need to see where it shows up. Consider the
following:
1.
𝑥 2 − 81 = 0
⟹ 𝑥 2 = 81
⟹ 𝑥 = ±9.
2.
𝑥 2 + 81 = 0
⟹ 𝑥 2 = −81
⟹ 𝑥 = ± −81
Complex Numbers
• 𝑥 = ± − 8 1 does not have a real solution.
• It has an “imaginary” solution.
• In order to define a complex number, we need to create a new variable.
• This new variable is “𝑖”, known as iota.
What is iota?
• "𝑖" is an imaginary number.
• Or a complex number.
• Or an unreal number.
• The terms are inter-changeable.
What is iota?
• It is a tool to solve an equation.
• It is defined as: 𝑖 = −1
Note that 𝑖 is representation of −1. It is not simplification of −1.
• So, following this definition we have:
𝑖 2 = −1,
𝑖3 = − 𝑖
and
𝑖 4 = 1.
What is a complex number?
• A complex number is a combination of imaginary and real
numbers.
• A complex number is a number that can be expressed in the form:
−1
𝒂 + 𝒃 𝒊,
Real part
Imaginary part
where 𝑎 and 𝑏 are real numbers, and 𝑖 represents the “imaginary
number”, satisfying the equation: 𝑖 2 = −1.
Observation
▪ All real numbers are
complex:
3 = 3 + 0𝑖.
▪ All imaginary numbers
are complex:
7𝑖 = 0 + 7𝑖.
Complex number
• A number such as 3𝑖 is a purely imaginary number.
• A number such as 6 is a purely real number.
• 6 + 3𝑖 is a complex number.
• 𝑧 = 𝑎 + 𝑏𝑖 is the general form of a complex number.
• The real part of a complex number 𝑧 is denoted by Re(𝑧) or ℜ(𝑧)
and the imaginary part of 𝑧 is denoted by Im(𝑧) or ℑ(𝑧).
• If 𝑧 = 𝑎 + 𝑏𝑖 = 6 – 4𝑖 then ℜ(𝑧) = 𝑎 = 6 and ℑ(𝑧) = 𝑏 = −4.
Complex Numbers and Their Properties
Equality
Complex numbers 𝑧1 = 𝑎1 + 𝑖𝑏1 and 𝑧2 = 𝑎2 + 𝑖𝑏2 are equal, i.e., 𝑧1 = 𝑧2 , if
𝑎1 = 𝑎2
and
𝑏1 = 𝑏2 .
In terms of the symbols 𝑅𝑒(𝑧) and 𝐼𝑚(𝑧), we say that 𝑧1 = 𝑧2 if
𝑅𝑒 𝑧1 = 𝑅𝑒 𝑧2
and
𝐼𝑚(𝑧1 ) = 𝐼𝑚(𝑧2 ).
Complex Conjugate
For a complex number 𝑧 = 𝑎 + 𝑏𝑖, the complex conjugate is given as:
𝑧ҧ = (𝑎 + 𝑏𝑖) = 𝑎 − 𝑏𝑖.
This is useful when we wish to carry out a division.
Note that:
(𝑧)ҧ = (𝑎 − 𝑏𝑖) = 𝑎 + 𝑏𝑖 = 𝑧.
Examples:
1.
𝑧 = 4 − 8𝑖 ⟹ 𝑧ҧ = 4 + 8𝑖.
2.
𝑧 = −2 + 3𝑖 ⟹ 𝑧ҧ = −2 − 3𝑖.
3.
𝑧 = −1 − 4𝑖 ⟹ 𝑧ҧ = −1 + 4𝑖.
Algebra of complex numbers
Simplify:
3𝑖 + 7 + (4𝑖 − 6).
3𝑖 + 7 + (4𝑖 − 6) = 7 − 6 + 𝑖 3 + 4 = 1 + 7𝑖.
Evaluate:
3𝑖 + 7 − (4𝑖 − 6).
3𝑖 + 7 − 4𝑖 − 6 = 7 + 6 + 𝑖 3 − 4 = 13 − 𝑖.
Algebra of complex numbers
Evaluate: (3𝑖) × (−4𝑖)
3𝑖 × −4𝑖
Simplify:
= −12𝑖 2
= 12.
3𝑖 + 7 (3𝑖 − 7).
3𝑖 + 7 3𝑖 − 7 = (3𝑖)2 − 7
Evaluate:
= −12 −1
2
= 9𝑖 2 − 49 = −9 − 49 = −58.
3𝑖 + 7 (−2𝑖 + 1).
3𝑖 + 7 −2𝑖 + 1 = −6𝑖 2 + 3𝑖 − 14𝑖 + 7 = 6 + 7 + 𝑖 3 − 14 = 13 − 11𝑖.
Algebra of complex numbers
Algebra of complex numbers
Useful Rules
Useful Rules
Practice Problems
1.
16 + 9𝑖 + 6 − 2𝑖
2.
8 + 3𝑖 − 16 − 21𝑖
3.
16 + 9𝑖 × 6 − 2𝑖
4.
8 + 3𝑖 ÷ 8 − 2𝑖
9 − 𝑖 4 + 2𝑖
5.
+
4
2+𝑖
Laws of Complex Numbers
Like real numbers, following laws are valid for complex numbers as well:
1. Commutative law:
𝑧1 + 𝑧2 = 𝑧2 + 𝑧1 .
𝑧1 𝑧2 = 𝑧2 𝑧1 .
2. Associative law:
𝑧1 + (𝑧2 + 𝑧3 ) = (𝑧1 + 𝑧2 ) + 𝑧3 .
𝑧1 (𝑧2 𝑧3 ) = (𝑧1 𝑧2 )𝑧3 .
3. Distributive law:
𝑧1 (𝑧2 + 𝑧3 ) = 𝑧1 𝑧2 + 𝑧1 𝑧3 .
Zero and Unity of Complex numbers
The zero in the complex number system is the number 0 + 0𝑖 and the unity is
1 + 0𝑖.The zero and unity are denoted by 0 and 1, respectively. The zero is the
additive identity in the complex number system since, for any complex
number 𝑧 = 𝑎 + 𝑖𝑏, we have 𝑧 + 0 = 𝑧. To see this, we use the definition of
addition:
𝑧 + 0 = (𝑎 + 𝑖𝑏) + (0 + 0𝑖) = (𝑎 + 0) + 𝑖(𝑏 + 0) = 𝑎 + 𝑖𝑏 = 𝑧.
Similarly, the unity is the multiplicative identity of the system since, for any
complex number 𝑧, we have:
𝑧 · 1 = 𝑧 · (1 + 0𝑖) = 𝑧.
Inverses of Complex numbers
In the complex number system, every number 𝑧 has a unique additive inverse.
As in the real number system, the additive inverse of 𝑧 = 𝑎 + 𝑖𝑏 is its negative,
− 𝑧, where −𝑧 = −𝑎 − 𝑖𝑏. For any complex number 𝑧, we have 𝑧 + (−𝑧) = 0.
Similarly, every nonzero complex number 𝑧 has a multiplicative inverse. For 𝑧
≠ 0 there exists one and only one nonzero complex number 𝑧 −1 such that
𝑧𝑧 −1 = 1. The multiplicative inverse 𝑧 −1 is the same as the reciprocal 1/𝑧.
Thus, for 𝑧 = 𝑎 + 𝑖𝑏, the multiplicative inverse is given as:
𝑧
−1
1
1
𝑎
𝑏
= =
= 2
−𝑖 2
.
2
2
𝑧 𝑎 + 𝑖𝑏 𝑎 + 𝑏
𝑎 +𝑏
Inverses of Complex numbers
Example: Find the reciprocal of 𝑧 = 2 − 3𝑖.
Solution: By the definition of division, we obtain:
1
1
=
𝑧 2 − 3𝑖
1
2 + 3𝑖
=
.
2 − 3𝑖 2 + 3𝑖
2 + 3𝑖
=
4+9
2
3
=
+ 𝑖.
13 13
Practice Questions
Book: A First Course in Complex Analysis with Applications by
Dennis G. Zill and Patrick D. Shanahan.
Chapter: 1
Exercise: 1.1
Q # 1 – 20, 25 – 34, 37 – 44.